A mortise-and-tenon joint is, as far as woodworking joints go, very strong and quite versatile. From stopped or through mortises to wedged or tusk tenons, you can have a lot of fun with these joints which can be as attractive as they are functional.
Some people use a dedicated tool (mortise chisel) in order to make their mortises, but this Instructable will show you how to make one using a few simple tools which you probably have around the house. I'll also give you a few options along the way, so you can see different ways to chop one.
After you're done here, consider this Instructable which gives you excellent tips on how to create the tenon that will fit into this mortise (and which also gives you some of the same advice contained here on the mortise).
Step 1: Layout
There are many factors that will determine the size of your tenon in relation to the board as a whole (in other words, whether you will have zero shoulders or four, or somewhere in between). So I'll leave that for another Instructable and simply say: Your mortise should be the same size as your tenon.
The only thing I'd add to that is that you'll have an easier time making your mortise if the mortise is the same width as your chisel, as you'll see in the next step.
Step 2: Chop Chop
Since the mortise here is 3/4" wide, I take my 3/4" chisel and start chopping away. In my very first chop (pic #1), notice the position of the chisel. The bevel is pointing towards the inside, and the blade is set in from the edge of the mortise. You can define the top and bottom of the mortise later.
Now that you've done that, move the chisel forward and give it a good strike with the mallet. Then pivot the chisel in the direction of the bevel; this will lever out the waste (pic #2). Continue to do this for the length of the mortise, keeping the bevel in the direction you are moving. See pic #3.
After you've done that, clean out the waste (pic #4). Depending on the hardness of the wood, you might be able to use your finger, or you may want to use your chisel or (in this case) a pocketknife. It doesn't have to look pretty - you're just clearing out the waste for a second pass. Pic #5 shows the status after the second pass. At this point, I take my chisel and pare the side walls of the mortise as well as better defining the top and bottom edges (pic #6).
A BRIEF ASIDE: Some tenons are intended to go all the way through the piece; they are called through tenons. If you were making a mortise for a through tenon, then at this point it would be a good idea to flip the board over and repeat these steps on the other side. Otherwise, you will eventually blow out through the other side of the board, causing tearout and other nasty things that will ruin the look of your piece. Make sure you are laying out your mortise precisely so the two sides meet in the middle!
This mortise is what is called a stopped (or stub) mortise, in that it does not go all the way through the board.
Step 3: Optional Drilling
Some people prefer to make their mortises by drilling first, and then removing the waste afterwards with a chisel. This definitely speeds up the process, but I would still advise doing the previous step of chisel work for two reasons. First, if your mortise will be visible in the finished product (like with a through mortise-and-tenon joint), then you'll want very nicely-defined edges, which is easier to do with a chisel. Secondly, having begun the mortise with the chisel, the drill bit now fits nicely and will not skitter across your workpiece.
In this case, I am making a stopped mortise, so I drop my drill bit in (pic #1) and have at it. I need to stop periodically to check the depth of my hole (pic #2) to make sure it's deep enough for the tenon. If possible, try to overlap your holes (pic #3), which will minimize your chisel work in the next step. If you have a drill press, then this is a very easy step, but if you are using a hand drill you will need to ensure you're not overlapping so much that the drill bit slips into a previous hole.
Once you're done drilling the holes (and if you didn't use a drill press), you might find it helpful to take the drill and go in at an angle to help ream out some of the waste (pic #4). Once you're done with that, you'll see something similar to pic #5. All the holes are at more or less the same depth. If you were making a through mortise, then hopefully the four walls will be mostly uniform the entire depth of the mortise.
Step 4: Final Cleanup
Brackets? Screws? Pocket holes? Nails? Pshhhh. Who needs 'em when you have mortise and tenon joinery?