The meaning is dependent on the context it is used in. It can also mean "I understand you", in the sense that "I got what you meant." It is a variant of "I get it", which you may also have heard.

It may also mean "I tricked/fooled you."

Sometimes people use it to signify that they have captured someone. this way of using the phrase is usually used in movies, but can be used in everyday conversation to signify that you now have a significant other. It is usually used this way when talking directly to them.

"Get" has many meanings, and we can only know the correct meaning from the context. Anthony has given two examples (capture and partnership... I'll assume these are exclusive :) ), and another is "I understood what you said."

I got you=面倒をみてあげます