"Have got" just means the same thing as "have", except for its applications in the perfect tense. Saying "I've got to go" means the same thing as "I have to go", and, similarly, saying "I've got a lot of friends" means "I have a lot of friends". "Gotta" is simply an informal combination of "got to". Note that you can't negate "have got" or use it in any tenses other than the present, either: you need to use "have".

It's important to know that "gotta" is virtually never written that way. It is always "got to." The only exception is in a quotation, where it is important to convey the casual nature of the speech.