The answer to this question will be deeply personal and will vary greatly from person to person.
In Matthew 5:30, we are told that we should cut off our right hands if the cause us to stumble. What this means is that we may have to give up things that we really enjoy because they are offensive to God. For some people, this might call them to give up certain behaviors. A person might decide that going out drinking with friends puts them in too many situations of temptation. But another person might have to go even further. They might have to give up people who are their close friends if they feel that those friends are a negative influence.
It is also possible for people to have to give up things that are not offensive to God. They might feel that they have gotten too caught up with these activities and do not have enough time to do what God wants. Let us say that you have a hobby that you really love, but which does not allow you to witness for God in any real way. You might have to give that hobby up so as to have more time to spend on the more important business of discipleship.
The specifics are very personal, but the idea is generally that we might have to give up things that are either offensive to God or which tend to take too much of our time away from his work.