One great benefit of vermicompost is that it's completely self-contained so you can compost if you're in an apartment. Vegetable and fruit scraps and shredded paper go in, and a few weeks later you have worm casings and compost which you can apply to your potted plants, give to friends, or put on plants in a city park.
The worm compost bin here is designed by is made by the NC Worm Farm. To build you will drill several holes in the bottom of one plastic tote then place that in the other bin. Make a spacer out of a lumber scrap, gravel, or whatever else to provide an inch or two between the two bins. Cut a four inch or so square out of the lid and attach the window screen material to the underside of the lid. Fill the bin with damp shredded paper and a bit of peat moss and your starter worms. You can usually find a pound or so of starter worms locally on Craigslist, but you can also buy them directly from the NC Worm Farm (or many other online sites), NC Worm Farm charges $29 for a pound of red wigglers delivered anywhere in the US and guaranteed to arrive alive.
From time to time you will notice some moisture collect in the bottom bin, this is the water runoff filled with residue from worm casings and compost. Called "worm tea" this stuff is some of the best natural fertilizer you can apply to plants.
If you're looking for an alternative to traditional aerobic compost bins consider giving vermicomposting a try.