You have a number of choices, here. A compost bin, box, tumbler, trench or heap - all will work, its a matter of personal preference. Let us look.

Compost Bin. This can be made out of just about anything; discarded 44gal. drums are good, provided they are clean of petro-chemicals; plastic cherry-bins are ideal. For the most part, it needs to be big enough that you could climb into it. Cut off the top and bottom - keep one of these for the lid. Make a number of "breathing-holes" around the shell - these can be either cut or drilled through. A plastic rubbish bin will do. Once again, cut off the bottom, make holes in the side - then turn it upside-down, so that the wider opening sits on the ground. The idea of no bottom, is to allow the insects, worms and micro-organisms which help degradation of your waste materials the freedom to migrate into and out of your compost.

Compost Box. This can be made out of anything you have to hand, be it old pallets, bricks, or spare lumber. Bales of hay work quite well - the hay ultimately becoming compost itself.

Plastic Composters. There are a number of plastic composters available, stand alone units and tumblers. These are made of a special grade plastic that "breathes". A tumbler, is a compost bin, turned on its side; and with the aid of a crank handle, is turned a quarter turn every couple of weeks or so - effectively mixing the compost for you. The is one draw-back with these, though - the tendency to turn it too often, or really crank-it-up; then you end up with this glutinous ball, not much good for anything.

Trenching. This is quite an old-fashioned way of composting, but certainly worth discussing. A trench is dug, about double spade-width, the soil left on the side of the trench. All your waste is placed at the start of the trench, without spreading it out - the soil left on the side, is then back-filled over the material you have just placed in there. Then, the next-time you have more material, it is placed at what is now the start of the trench; and back-filled again. When you have run out of trench - start a new one. This is actually a good way to revitalise vegetable gardens, ordinarily made in bed-rows. An entire row is dug out and becomes the trench, filled with organic material, back-filled; and left to its own devices for about 6 months - then turned over and made ready for planting again.

Compost Heaps. A compost heap, is just that - a heap of compost material. However, the whole process will work much better, if it is managed a bit. By this I mean "caging" it. Using 4 to 6 stakes or pickets hammered into the ground, in a vague circle; chicken-wire, sparrow-wire, or arc-mesh is then formed around them, creating an open-ended cage. Large sheets of cardboard can be used; and will eventually decompose, too. Black sheet plastic can also be used as a surround, but it tends to deteriorate and tear, fairly quickly - then you have to get rid of it some way. Back to the dump, oh,dear.

The basic idea of all these designs can be seen here.
Worm Farms. An alternative to composting in the traditional sense is vermi-composting or worm-farms. These use a special kind of worm to break down kitchen scraps producing a fine compost-like material from their casts, or leavings; and a nutrient filled liquid plant food which is ideal for feeding indoor pot plants. If you have a greenhouse, or have a lot of houseplants then a worm-farm may be the best choice for the disposal of household waste. These can be constructed by the home gardener, or store bought. There are some marvelous small, neat units quite suitable for people living in apartments, too.