Depending on the context it can mean several things:
"I've got it." in the sense that "I can handle this." or "I will do this." This basically means the person will do whatever 'it' is. If parents needed to pick up the kids from school, the dad could say "I got it.", meaning that he will pick them up.
"I got it." or "I get it." can be used to acknowledge something. Used in this way, it can, but does not always, have a bit of a negative tone. If someone was explaining something, you could say, "I got it." to mean that you understand what they are saying. If they are being overly specific or being too simple to the point that you feel like they are being condescending, or looking down on you, then saying "I get it." with a harsher tone implies that you don't need them to tell you it anymore.
"I've got it." or more commonly, "I've still got it." is an expression to mean that you can do something well. This is usually seen in 'picking up' women, where the guy says this to mean that he still knows how to woo a woman. In a non-dating situation, you could play a game and win, and say "I've still got it." to mean that you still are skilled in playing that game.