Pocket gophers get their name from an anatomical oddity that is found in their cheeks: a fur-lined pouch on either side of their mouths into which they stuff food for later consumption. They range from 5-12 inches in length and can weigh up to two pounds.
"True" Gopher
Columbian Ground Squirrel
Gopher Burrowing
Gopher Kill Trap

I have lived in the Gopher State my entire life and had always thought of a gopher as something like a ground squirrel, woodchuck, or prairie dog. Alas, I was completely mistaken. The one true gopher, the pocket gopher (family Geomyidae), is an elusive and destructive creature ranging in size from thirteen to thirty-six centimeters in length. There are over thirty species in the Western hemisphere, including thirteen in the United States. Most of us have probably never seen the animal alive, but we would have seen the ripped up gardens, piles of dirt in the lawn, and damage to building foundations and golf courses. These are all good reasons to hate gophers, even though they are just doing what comes naturally to them.
Look for signs. As I mentioned above, gophers are elusive, and unless you start invading their tunnels and forcing them out or decide to sit quietly in one place for a few hours, you probably won't see one either. But what you will see are holes and distinctive piles of dirt. They tend to toss all of their dirt out to one side, like a dog digging a hole, whereas a mole builds the entrance in a volcano shape. Take action once you have spotted the problem; do something about it! This is important because it doesn't take long for gophers to breed, and then you will have a real problem. This call for vigilance applies to just about any pest problem.
Traps are one way to kill gophers. The classic gopher trap you see in hardware/garden stores are classic for a reason: it works really well (that is, if you can figure out how to set it and then get it into the hole without setting it off on your hand). What appears to be a confusing knot of twisted metal is actually known as a Macabee or Death Clutch trap, and if deployed correctly, it works like a charm. Remember, this is a trap made to be used in a tunnel; it's not like a mouse trap—the trip rod doesn't hold the jaws open. I made this mistake the first time I picked one up. It's actually quite simple, just different.
Using poisons to kill gophers. I normally don't recommend poisons because of the risk to other animals. But, because gophers spend most of their lives underground, this risk is slightly lower. The poisoned bait is usually some kind of strychnine-soaked grain that you need to get into the path of your target. This is done by inserting a hollow probe of some kind into a known tunnel and then feeding the bait down the tube. There are actual devices made for this purpose, and if you have a real problem, it might be a good investment. On the other hand, a metal or PVC tube with a funnel on top will do the trick and cost you practically nothing.
Gas gophers out with fumigants. There are several options here: smoke sticks, aluminum phosphide pellets, poison gas, or exhaust piped in from a fuel burning engine (preferably one without a catalytic converter). The use of these methods is a point of contention in pest-control circles. The issue is that gophers can have hundreds of feet of tunnels, making it difficult to gas them out. If you have a good idea of how the tunnel system is laid out, a strategy can be formulated. Your best bet is to block all exits, especially from peripheral tunnels, or lay traps at one end of the tunnel system and then start your gas attack at the other, working your way towards the traps.
Guns will work, but you will need patience. The most important thing to remember when hunting anything is gun safety. Don't shoot towards people, houses, or solid surfaces. Look for holes that have a source of vegetation nearby—that is why your gopher friend is coming out of the safety of its tunnel. If the hole is recently dug but capped, it means that the gopher is done in that area for the time being and has moved on somewhere else. You might try uncapping the hole and waiting nearby for the gopher to make a repair. They can't see or hear very well, but they will be able to smell you, so keep downwind if you can.

Natural borders. Gophers prefer plants with a single tap root, like carrots or alfalfa. In an agricultural setting, creating a border with a grassy plant (like a grain) around your more valuable crop creates an inhospitable environment for the gopher.

Crop rotation. Keep them guessing. If they have moved into an area, there is a reason—they like what you are offering. Plant something less tempting to the gopher (like a grassy cover crop) for a season, and it might abandon its burrow.

Demolition. This method is for those of us who enjoy digging, although I don't know anyone who fits that description. I can see doing it out of curiosity or for scientific purposes. Just be sure to plan ahead as to what you will do if you should find yourself face to face with a gopher.