Last summer I adopted my very first cat :) Back when the weather was a little warmer, we didn't mind having the doors between the garage and the rest of the house open all the time, so miss whiskers could go wherever she pleased. However, as it's winter now, we keep the doors closed to avoid cooling down the entire house.

To allow the little lady to keep roaming around freely, a pet flap is in order! Since this flap is meant for indoor use, it doesn't have to be weatherproof, so I figured.. why not make one :) Let's get started!

Step 1: Tools and Materials

While I used somewhat more specialized woodworking tools (e.g. a bandsaw), this project can be made just as well using basic tools and materials that you can find in just about any hardware store. Something to note as well is that the measurements I'm using here are suitable for small/medium-sized cats. To adjust the pet flap's measurements to your pet, you can browse for a suitable commercial pet flap and use the dimensions listed on their website as a reference.

Tools:

Bandsaw (alternatives: backsaw and/or handsaw)
Scroll saw (alternative: coping saw)
Jigsaw
Hacksaw
Hammer
Small wood files
Small chisel
Drill (with 2 and 6 mm drill bits for wood)
Drill press (optional)
Screwdriver
Drawing and measuring tools: pencil, eraser, utility knife, a ruler and a try square

Materials:

Small sheet of wood, at least 1.5 cm thick (I used pine wood)
Small sheet of acrylic (I cut a piece of 16 x 16.3 x 0.4 cm)
4 small wood screws
8 small nails
Double-sided tape (alternative: adhesive spray)
All purpose glue (to attach acrylic to wood)
Wood glue
Sandpaper (I only used 100 grit)

Step 2: Making Flap Decorations

The pet flap we're going to make essentially uses a piece of acrylic as the flap; it has a wooden hinge, and two decorative thin sheets of wood with a paw symbol attached to the front and back side of the flap. I first started with making the decorations (shown in the first picture):

From your sheet of wood, cut a 13 x 13 cm piece.
Next, we need to resaw this piece into thin slices; we only need two of them. Mine were about 2 mm thick. (Shown in the second picture. You'll need a bandsaw for this step. An alternative would be to directly buy a sheet of wood veneer, or to use a different material like a thin sheet of plastic.)
On a piece of paper, print or draw whatever decorative shape you'd like to cut out from the two slices. I chose a picture of a paw, but e.g. a silhouette of your pet would be nice too :)
To make sure the cutout looks the same in both slices, we need to temporarily stick them together. I used thin strips of double-sided tape for this (shown in the third picture).
Put the paper with your shape on top of the two slices, and carefully cut out the shape from the paper with a utility knife (fourth picture). The goal here is to carve the shape directly onto the wood.
Once the shape is carved onto the wood, trace over the shape again with a pencil so you can see it a lot better (fifth picture).
Drill a few starter holes into the parts that need to be cut out. Be sure to provide enough support underneath the wood to avoid breaking it while drilling.
Use a scroll saw or coping saw to cut the shape out of the wood (sixth picture).
Carefully pry apart your two slices again and remove the double-sided tape.
Finally, use sandpaper and/or some small wood files to clean up the edges. Optionally, you can also use the sandpaper to add a subtle rounding effect to the edges.

Notes:

It's best to use quite thin slices of wood to make these decorations: The thicker the wood, the more weight you add to the flap and the more force your pet will need to open it. That said, the flap I made is a bit heavier than your average pet flap, but my (1-year-old) cat opens it with ease.

Step 3: Making a Hinge

The hinge of the pet flap (first picture) consists of a top and bottom part, where the top part will be attached to the door, and the bottom part will be attached to the acrylic flap.

Top hinge part: (measurements shown in the second picture)

First cut a 16 x D x 1.5 cm block of wood, where D is the thickness of the door. (In my case D=4 cm.)
From this block, cut away two 16 x (D-1)/2 x 1 cm blocks, to make the T-shape shown in the third picture.
On both sides of the T-shape, drill a 5 cm deep hole with a 6 mm (or smaller) drill bit.
Now cut two 1,5 cm-sized notches by making multiple passes with the bandsaw (fourth picture). Alternatively you can also use a coping saw here, or make a few passes with a hand saw.
Using some sandpaper, round over the edges that will touch the bottom hinge part as much as possible. If you don't, the hinge won't be able to rotate :)

Bottom hinge part: (measurements shown in the fifth picture)

Cut a 16 x 1 x 2 cm block of wood.
Drill two 5 cm deep holes on both sides of the block.
Now cut away chunks such that this block will interlock with the top hinge part: First cut two 1,5 cm-sized chunks off both ends (sixth picture). To cut away the middle chunk, first make some room by making multiple passes with the bandsaw (seventh picture).
Using the bandsaw (or a backsaw), cut a groove for the acrylic flap to fit into (eighth picture). I've cut mine 3 mm deep, and 4 mm wide (i.e. the thickness of my sheet of acrylic).
Check whether the groove fits snugly onto your acrylic sheet. If not, you can still widen the groove using some wood files and/or chisels.
Using sandpaper, round over the edges that will touch the top hinge part as much as possible.

Notes:

To drill a hole of a certain depth, you can add a marker to your drill bit by simply attaching a piece of masking tape at the desired depth.
If you have a drill press, it's best to use it here when drilling the holes. If you don't (as is the case for me), clamp down your piece and drill vertically for best eyeballing results :) It still won't be all that accurate, but that's okay; we can still make corrections afterwards.

Step 4: Assembling the Pet Flap

In this step we'll assemble our pieces to create the entire pet flap (first picture):

Using the bandsaw, cut a 16 x 16.3 cm piece of acrylic, which will become the actual flap of our pet flap (second picture). Alternatively .. I've never tried this, but it should also work using a hacksaw.
Using all-purpose glue, glue the two decorations onto the acrylic flap, one at a time (third and fourth picture). To add clamping force while the glue is drying, stack a few books, boxes, .. on top.
Now glue the acrylic flap into the groove of the bottom hinge part. You can just let it dry by itself if you already have a snug fit; no clamps needed.
We still need something to connect the top and bottom hinge parts. I used a hacksaw to cut two 5 cm-long cylinders from a metal rod that I scavenged from an old scanner (fifth picture). You may need to clean up your cuts with some sandpaper or files.
Insert the two cylinders and test whether your hinge works smoothly, and whether the flap is level relative to the top hinge part. If not, you can use some small wood files (or your drill) to carefully make small adjustments to the holes you've drilled.

Notes:

It's okay that the hinge's cylinders can still slide/fall out at this point. Once the pet flap is installed in the door, they won't be able to get out.
Keep in mind the diameter of the two cylinders that join the top and bottom hinge parts. (Mine have a 3 mm diameter.) The smaller the diameter, the more room the two hinge parts will have to rotate, but the bigger the gap will be between the two hinge parts. You'll want to keep this gap small; otherwise the pet flap won't work very well as thermal insulation.
If needed, you can use a small chisel to chip away a little more room in the notches of the top hinge part, so the hinge can rotate freely.
To make sure the decorations are positioned correctly, I made a quick L-shape on the bandsaw to help out (fourth picture).

Step 5: Making a Door Frame

We now have our pet flap, but we still need to attach it to the door. If you have a solid wooden door, you can just cut your hole in the door, and directly attach the pet flap to it. If you have a hollow core door (filled with a cardboard pattern inside), we'll first need to make a door frame (first picture):

Cutting a hole:

It's not really necessary, but I first removed the door from its hinges and laid it flat on my workbench. It's a bit more convenient to work with like this.
Using a pencil, ruler and try square, mark the rectangle you want to cut out of the door. This rectangle should be 20 x 17.2 cm, which is the size of the entire assembly, including the door frame. Note that the bottom side of the rectangle should roughly correspond to the height of your pet's belly, measured from the floor (when standing in a neutral position). This should give your critter easy access when walking through the flap.
Inside each corner of the rectangle, drill a starter hole. From these starter holes, use a jigsaw to cut the rectangle out of the door (third picture). It's okay if the cut looks a little rough; it will be covered up by the door frame anyway.

Inner part of the door frame: (measurements shown in the fourth picture)

Cut two 19 x 0.5 x D cm pieces of wood, and two 17.2 x 0.5 x D cm pieces (where D is the thickness of the door).
Use wood glue to make a 20 x 17.2 cm rectangular shape out of the four pieces, as shown in the fifth picture. Be sure to test that the corners are square when the glue dries.
Slide the rectangular shape into the door's hole. If the hole isn't quite big enough, use a rasp to make small adjustments until you get a snug fit.
If the shape sticks out of the door a bit too much, you can make adjustments with a chisel (sixth picture).

Outer parts of the door frame: (measurements shown in the seventh picture)

The outer parts of the door frame essentially look like two picture frames that cover both sides of the door, and will prevent the inner part from moving around. To make these picture frames, first cut four 22 x 1.5 x 0.3 cm pieces, each with 45° cuts on both ends. (If you're cutting these by hand, you can use a mitrebox to make the 45° angled cuts.)
Likewise, cut four 19.2 x 1.5 x 0.3 cm pieces, again with 45° cuts on both ends.
Using wood glue, attach the pieces to make two 22 x 19.2 cm picture frames (eighth picture). Be sure to test that the corners are square while the glue dries. I also added some weights on top of the corners just to have a little bit of clamping force.
Once the glue has dried, and with the inner part of the door frame still in place, the picture frames can be attached to the door with a few small nails (first picture). Because these picture frames are rather thin, they're also very fragile when hammering in nails. If you have a teeny-tiny drill bit laying around, it's a good idea to pre-drill the holes before hammering in the nails.

Notes:

The outer parts of the door frame are only 0.3 cm thin purely because I like how it looks; you can make'em thicker just as well :)
For additional strength, you can glue (one of) the picture frames to the door frame's inner part. Likewise you can also glue the picture frames to the door, in addition to the nails. I haven't done any of this so I can still easily take the entire pet flap apart in case I ever need to make any repairs. (The pet flap is holding out fine without the glue so far.)
The inner part of the door frame could also be made stronger by creating e.g. a box joint between the pieces. Then again, this inner part is already supported on all four sides by the door itself, so it's not really necessary.
Note that the door frame's measurements provide 1 mm of slack between the pet flap and the door frame, to make sure the flap can move freely.
When attaching the outer parts of the door frame with nails, it's best to stay away from any knots in the wood grain! I managed to break my picture frames twice when attempting to hammer a nail near them. I ended up glueing the pieces back together, hiding the holes I made with some saw dust + wood glue, then adding the nails elsewhere, avoiding any knots.

Step 6: Installing the Pet Flap

Almost there! The final step is to install our pet flap into the door frame:

If you've removed the door when making the door frame, you can now reinstall the door.
Temporarily remove the cylinders from the hinge to retrieve the top hinge part. Now drill some pilot holes in the top hinge part (second picture).
Drive a screw into each pilot hole, just so the tip of each screw pokes out on the other side of the top hinge part (third picture).
Hold the top hinge part in the position where you want to mount it in the door frame, just so the tips of the screws will leave a mark in the door frame. (If possible, drill some pilot holes in the door frame as well.)
Reconnect the top and bottom hinge part and reinsert the two cylinders.
Finally, using a screwdriver, screw the pet flap onto the door frame (fourth picture).
All done; the pet flap is ready for action! If all is well, the pet flap can move freely without touching the door frame. Time to put your flap to the ultimate test and see what your pet thinks of your handiwork :)

Notes:

In case the flap is scraping against the door frame, you can still take it apart and file/sand down the acrylic flap until it can move freely.
If needed, you can always add some sort of bolt lock to be able to lock the pet flap in one or both directions.
Note that I haven't added any wood finish to this project. I figured cats'd prefer the scent and feel of natural wood .. well, I do :) I have a few other projects without any wood finish and they're holding up fine; the wood colour only became slightly darker over the years.