Aquaponics is the commingling of Aquaculture, a.k.a. fish farming, and Hydroponics, growing plants/crops without soil. This instructable will show you a simple method of growing both fish and plants, using easily obtained and/or household items, for enjoyment and beauty or for consumption.
Why should you care?
Aquaponics (AP) is easy and has been used for thousands of years (from Chinese rice paddys to the Aztec water gardens), can be practiced on any scale, consumes approximately 1% of the water in a traditional growing system, and is a great way to learn about closed-loop systems and ecology.
AP can help feed families in dire need of nutrition, with agricultural crops; but also with much needed proteins in the form of fish. This example isn't exactly what a poor family in a third-world country might put together, it is exactly as stated; an example.
The entire earth is (more or less) a giant aquaponic lab; but we're going to start small, in the living room.

Step 1: What You Need

I'm using a simplified "S&S Aqua Farm" version of Aquaponics. I'll go over some different thoughts on AP and different styles of systems towards the end of this "ible."

Tools needed:
Dremel
Scissors

Materials:
Rubbermaid Roughtote 18-gallon storage bin.
~18-gallons of water
~8-gallon "under bed" style tote.
~8-gallons of "pea gravel"
~4-ft of 1/2" tubing
2-liter bottle
End cap for tubing
Twine
~200-gallon/hr water pump with control valve (less is probably fine with a system this small)
On/Off timer with 15-minute intervals
Freshwater aquarium water tester (strips or kit)
Fish (I used 3 goldfish and a crayfish)
Seeds for planting (Fruiting or non-fruiting, lettuces, wheatgrass, it's all good)
Short stand or table that can hold ~80lbs or more.

If you had to buy everything for this as new, it would probably cost ~$30US including the fish and seeds but excluding the stand/table and test strips. As it is I had everything on hand except the pea gravel, so this cost me about $0.75 to make.

Options (not necessary for the purpose of this Instructable, but very valuable additions to AP in general):
Aquarium heater
Thermometer
Uninterruptable Power Supply (battery backup)
Grow lights, CFLs work fine (if growing away from sunlight)

Step 2: Cleaning Gravel

This is the true grit, the right of passage for any AP system keeper. You must now clean your gravel. It is very rough going, i used a big colander and several batches of gravel under a hose, and more in the bathtub, and it still wasn't clean when i added it to the grow bed.
You can avoid all this by using LECA, expanded clay aggregate, a.k.a. Hydroton, instead.
Hydroton is also much lighter than pea gravel, if you can afford it, definately go for the Hydroton.
Otherwise, GET TO SCRUBBING!!!

Step 3: Put It Together

Basically, you want the growbed, filled with gravel to sit partially above your fish. For 15 minutes, the pump pushes water above the tank and into the grow bed. The grow bed has holes on the bottom corner that sits above the fish tank, emptying back into the tank.
We achieve this by using the Dremel to drill 16 1/16" holes about 1/2" apart on the bottom of the under-bed tote(grow bed). We also drill 8 1/4" holes near the top of the grow-bed to save overfilling the bed and flooding the living room! Hold the Dremel in a way that you are drilling from the middle of side that is over the fish tank, if you drill from the center the water will pour out of the grow bed towards the middle, also saving you from flooding. Also, drill at least 4 1/8" holes on the upper rim of both sides that are perpendicular to the side over the tank, directly across from each other; these will help secure the tubing later.
Place the roughtote on the floor, next to the stand that holds the grow bed and fill it partially with water. Hook the tubing up to the water pump and place the pump inside the tank, with the other end of the tubing securely in the grow bed. Make sure the grow bed is hanging over the tank in a way that the water will empty back into the tank, and plug-in the pump. Adjust the pump's control valve to slow water uptake as necessary. When you think you've got it close to perfect (water rises in the grow bed to the 1/4" holes, but not past), unplug the pump.
Fill the 2L bottle with water and cap it; place this on the side of the grow bed where the drainage holes are drilled. Then fill the grow bed with your washed (and re-washed:) gravel. The bottle is there to protect your drainage holes from the root structures of the plants you'll be growing. Those roots will plug the holes and you'll have another potential flood in the room.
Place tubing over the gravel. Using the scissors, cut the tubing at the far end of the grow bed and use the end cap to, uh, cap it. Drill 5 or 6 1/8" holes all the way through the tubing, spaced along the length of the tube. Using the twine (will eventually rot, break, and need replacing), tie-down the tubing so that it doesn't move when there is water pressure.
Time to test the pressure on that pump again. Adjust it as necessary.
Take the lid from the roughtote and cut enough space out of it to allow your water to fall through, but retain the handles and connect the lid to the roughtote for extra support. I also use the cut-out portion laid across the top as a splash guard.
Congratulations! You're done with the construction part of this "ible."
However, DO NOT PUT FISH IN IT YET!!! Doing so will almost certainly kill them.
Next step will get into some specifics on how this whole thing works...

Step 4: How AP Works

Aquaponics works in a similar fashion to freshwater aquariums with "biological" filters. We're taking the bio-filter concept and expanding it. Similar to a new aquarium, a new AP system needs to "cycle." Cycling is the means by which we start the "Nitrogen Cycle" (a.k.a. nitrification process, start-up cycle, biological cycle). This process adds ammonia(NH3) or ammonium(NH4) to the water in order to start the colonization of beneficial bacteria in the filter. In AP, the aquarium can be any tank that holds water and fish and the filter is a grow bed where crops are grown. Two types of bacteria colonies emerge, Nitrosomonas and Nitrospira. Nitrosomonas breaks ammonia down into nitratis, through oxidation, effectively cleaning the ammonia from the water. This would be a problem, as nitrites are just as poisonous to your fish as ammonia, if not for the development of the second beneficial bacteria. Nitrospira form and turn the nitrites into nitrates, again cleaning the water. What is left is easily consumed by growing flora, completing the cycle, and returning clean, safe water to the resevoir (fish tank/pond).
The cycling process can take anywhere from 2-3 days to 2 months (I've never experienced anywhere near that long, but supposedly it can happen). Test your water every couple of days for PH levels, and most importantly, for Nitrite and Nitrate levels. You will initially see a spike in Nitrites, then a spike in Nitrates as the Nitrites decline. When the Nitrates begin to rise it's time to add some flora to your grow bed. Your plants will happily consume the Nitrates through osmosis and the cycle is complete. There are several ways to begin cycling an AP system:
1. Put fish food or raw shrimp or other grocery store fish in the system, they'll begin breaking down into ammonia (or ammonium if your PH is low [~6 or under]), which will start the forming of bacteria.
2. Buy some 100% Ammonia at the store and add a bit (depending on the size of your system) every couple days.
3. Pee in it (probably not a good idea if you're on medications, narcotics, or are a heavy boozer).

Notes on speeding up the process: Using biological addatives such as "Colonize" or "Stress-Zyme", or filtration material or gravel from a mature AP/aquarium that is already colonized can greatly decrease cycling time. Increasing water temperature to a tepid bath (82-84F) and keeping the water pump on 24/7 is probably a good idea as well.

You also may have noticed that I'm not using any air stones and air pumps. For this style of aquaponics it's unnecessary unless you're stocking a LOT of fish. In a flood and drain system such as mine, the grow bed acts as a piston pushing old air up and out of the grow bed, and pulling fresh air in when the water level drops. This brings plenty of oxygen to the root systems of the plants. The water splashing back down to the fish tank incorporates into the water all the air your fish will need. So there is no need for an air pump.

You are now ready for the final step:

Step 5: Fish!

With this small setup I didn't want to go overboard with my fish. I stuck to 3 goldies and 1 crayfish. It's enough effluence to keep the small grow bed in nutrients, and enough grow bed to completely clean the water. I have my timer set so the pump is on for 15 minutes, then off for 15 minutes all day long. It's a perfect balance for what I have setup. That doesn't mean that there are hard and fast rules as to how to stock, how, or even IF to flood your growbed, or how to empty it. This is a simple example of this style of AP.
For this type (S&S Aqua Farm style) AP system, the stocking density and growbed to tank ratios are generally quote as 2:1 grow bed to tank and 1 pound of fish per 2 gallons of water (Note that this is an extreme in intensive aquaponics that is not recommended without extreme care and air pumps and stones). That doesn't sound to me like it would make for happy fish; but if you're in need a LOT of fish and crops, those are the limits.
I usually plant half of the grow bed at a time, so that when i harvest one side, the other side is still cleaning the water for me. I believe this is baby arugula i have growing in this pic.
There are other styles of aquaponics, some can be seen here at Instructables, others can be found all over the web. My personal favorite site for AP is http://backyardaquaponics.com. They have a wonderful forum of immensely knowledgeable people. Please visit there and buy the book, if you're into that sort of thing!

This brings us to the end of this instructable. I hope you enjoyed and will give AP a try! Expect a more elaborate "ible" on a much larger AP system in the near future...