There is a huge range of compost bins available, from the luxury to the utility to the home-made. They all have their merits and their disadvantages; but the incredible choice makes it pretty sure that one of them will suit you.
the Daleks - these are great, entry-point bins. They have their problems: the lids blow off; they tilt (under pressure from bad loading, dogs and weather); they have no base so rodents can get in - but they are cheap, effective, and inoffensive: they come in green, grey and black; totally round; oval or square. Usually around 200 litres or 300 litres. They are often discounted by Councils and offer a good introduction to composting (note that the delivery charge is seldom per bin - so get together with friends). They work well for leaf mould too, although they are perhaps an extravagant option for that purpose.
the Green Johannas - these are great bins; dearer than the Daleks, but they have a perforated base, are very stable and secure. They are a b****r to empty, and it is easier to dismantle them at the second tier than to try to shovel out of the doorway, but they are sturdy, contained and inoffensive. Often discounted by Councils, they are good value, trust me!
Wooden bins - these range from simple slot-together structures to convincing "bee-hives." They are the ideal choice where appearance is important, and they are flexible. Dearer than the dalek (and not often subsidised by councils), but if appearance is important, they are invaluable. A very basic wooden bin is cheap (and easily made from relatively cheap materials.
For additional cost, wooden bins can be extremely beautiful and add a new dimension to your garden.
And with ingenuity, a whole composting system can be established relatively easily.
Or do it the easy (but expensive) way with these modular plastic kits:
Tumblers - range from basic to state-of-the-art. They are intriguing, and no doubt in the right situation, they deliver. We have not yet found the right situation....
Up-and-over can be hard to turn; the cranked variety are expensive and depend upon batch processing; the micro-tumbers are cute, REALLY EXPENSIVE, and definitely need batch processing.
(Good grief! I've just discovered that they now supply this in shocking PINK!)
Digesters are entirely different, they are devised to actively digest organic matter, not simply leave it to decay.
Food digesters such as Green Cone are designed to exclude vermin, so are suitable for all kitchen waste. They harness solar power (through the double skin of the cone) to create the right conditions for bacteria to digest the waste materials. The 'basket' buried in the ground under the cone allows digested material to leach into the soil or be carried away by worms.
Bokashi bins (more or less under the kitchen sink) introduce the right sort of microbes to do it all for you. It's a fermenting process, with a sharp smell.
The Bokashi environment is anaerobic - without oxygen.
Anaerobic digestion is becoming more popular and can be undertaken on more or less any scale.
DIY bins have an infinite number of possibilities, and various materials can be used:
Wormeries are a specialist composting technique majoring on the power of worms to digest waste on your behalf. Don't knock it, until you have tried it! Why wouldn't you sit back and let an invertebrate do you work for you??