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There are two kinds of memory in a computer, RAM and Storage(hard drive) Memory.

RAM is what your computer uses to load programs, play video, music, etc. Think of it like a handyman's work bench. The more room he has, the more projects he can work on simultaneously and the faster he can access each of them. More is always better when it comes to RAM, so don't try to cut costs on this feature but don't break the bank going to large amounts either. 4GB should be plenty.
Storage Memory is what your hard drive is. It's where all the stuff you install and save get stored. If you plan on using your laptop for music, video, games, etc. you'll want as big a hard drive as you can afford. If your laptop is going to be used just for casual web browsing, e-mailing, IM'ing, essay-writing, etc. then this is definitely a feature you can minimize to save some $$. I would advise getting at least a 100GB hard drive as Windows, Microsoft Office, and other essential programs can really add up in memory usage over time.
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This again depends on your planned usage. If you want to play movies and games you're going to need a processor that's beefy enough to handle it. But if you're just doing casual tasks like web browsing, e-mailing, etc. then this is another feature you can cut costs on to save lots of $$. Don't go lower than 1.6GHz though, this should be your bare minimum.