I definitely agree with what Fett101 said. You should go to a store to try out the computers for yourself, a sort-of “try before you buy” aspect. I know that at an Apple store, they have tons of computers out for you to tap into. (Not to mention laptops.) That’s the only sure-fire way to find one that works for you.
I, myself, would recommend a MacBook Pro for intensive graphic design and animation. I wouldn’t suggest buying RAM from Apple, though. My suggestion is to buy a standard MacBook Pro, if you can afford it, and then upgrading the memory yourself by getting it from a 3rd party site. I recommend www.ramjet.com . Cheap memory. However, a MacBook would work just as well if that’s all you can afford right now.
I’m not such a big PC user, so I can’t recommend a good PC to try out. Like I said before, you should go to a computer store and try out the laptops for yourself. Start at an Apple store and work your way over to a… “PC” store like BestBuy. They almost always have some sitting out for you to try, and if you speak with a salesmen about exactly what you want, he/she should be able to help out.
Again, I’m not certain what you’re planning to do with a computer. I mean, if all else fails, I would recommend a gaming PC (laptop). You’d be able to play games and take advantage of the extra memory for CG animation and design. I would tell you to look at Alienware, but their setups are pretty expensive.
I’m a Mac user, so I’ll naturally want to go with a Mac. But I realize they can be expense. For a baseline, I’d say the lowest MacBook you can get will probably get you where you need to go… just a little slower.
By the way, can I ask what your budget is for something like this? Is your mom only wanting to spend like $1,500? This would help me in giving you a more accurate response, but again, I’m not a professional. Like I said before, just go in an Apple store/PC store and talk to a salesmen if I’m not helping you out at all. 
I hope at least some of this giant text wall helped you out. 