Once you get into Scrum, it isn't hard to remember the terms. A 'story' is just a feature to implement. It's called that because it's supposed to be seen from the eye of the customer, rather than the developer.
A Sprint is a timeframe for completing a bunch of stories. Usually 2 weeks, but some find shorter sprints are better, and others need longer ones.
As for the question: Yes, it was WAY better than waterfall for us. It forced everyone to get organized. Management could no longer be ignorant about what they were doing to the developers and testers on a daily basis. Instead, they were forced to actually put things on a timeline and decide what was more important, instead of just getting upset when everything didn't magically happen.
As a bonus, though, they gained the ability to estimate when things would be implemented, and changes to the system were rolled out every couple weeks, instead of going months between releases, and having a ton of bugs when it finally did launch.