This is going to be kind of hard to explain, but I'll try my best. Whenever you become a vampire, your special ability is nothing but a stronger version of a power that you already had when you were human. If you've read Breaking Dawn, you know that once Bella becomes a vampire, her special ability is a mind shield. That means that she can place a mental shield over herself and other people so that special abilities that deal with the mind (i.e., mind reading) won't work on them for as long as they have the shield over them.
Bella's power is a shield against mental abilities. Some of the characters, like Bella, Alice and Jasper, had their powers showing weakly while they were still human. Bella was already immune to Edward and Aro's mind reading and Jane's torture, but she couldn't extend her shield yet or lower it.