I highly suggest upgrading your computer.
But barring that, yeah, pre-Pro versions of Adobe Premiere should work.
And to answer your other question: Premiere is what's known as an NLE (<b>N</b>on-<b>L</b>inear <b>E</b>ditor), which means that it allows you to arrange video and audio clips in a multi-track setting and without having to place clips strictly from start to finish (therefore Windows Movie Maker, which does require clips to be placed linearly, is not an NLE; I believe it's simply known as a linear editor). As such, Premiere competes with products such as Sony Vegas, Magix Movie Edit, and Apple's Final Cut.
It's not as well suited for effects work and composition as, say, Adobe After Effects, but you can still do a lot with it.