How do you deactivate a mine?
You blow it up. Usually the military doesn't risk lives trying to defuse a mine. They just dig down next to it, place a block of C4 down there, then blow the damn thing up. Problem solved. Another similar method is to use a bangalor torpedo, which is basically a small explosive on the end of a stick; you run the stick right up next to or under the mine, then blow up the explosive. This method is only used in the heat of battle, however.
There is also a quick way to get a path through a minefield. There is a certain type of machine that can shoot basically a rope with a bunch of c4 attatched to it accross the field. Then the C4 is all blown up, clearing a path through the minefield. Just to be safe though, an armored bulldozer is driven through the path to push aside or detonate any surviving mines.
What's the differences between mine types, and could you give me some names of different types?
There is a $hit load of mine types, but there are a few that are the most common. Go to the chain of command's website, and download their mine addon. It includes some of the most common mine types used. Even if you don't use these in your mission, you can still see how the mines work. Make sure to look at the notes section of the example missions.
How deep are mines dug?
Depends on the mine; some are just a couple inches down, some are a foot or more deep.