A worm bin must be:
Worms used for vermicomposting are surface-dwellers, or epigeic worms, found under loose leaf litter or under manure piles. A bin needs to mimic those conditions. In the wild, worms can just crawl away if the conditions are bad, but in a bin, worms are at our mercy. It's up to us to keep conditions perfect for them.
It's best to keep your bin indoors, as worms prefer the same temperature we do- about 70F. Cold and heat will kill worms- don't go below 40F or above 90F. Standing water or too-dry conditions can also kill worms.
If you do keep a bin outdoors, keep it under a shelter, like a garage or shed. It needs to be in the shade- never let a bin sit in the sun. If you live in a cold climate, the bin must be insulated. It can be tricky to strike a balance between good airflow and functional insulation. Temperate climates can keep bins outdoors much more easily than northern climates.
There are several excellent commercial bins out there. My first bin was the Can 'O Worms. It is a multi-tiered bin, well-aerated, and not at all bad-looking. There are 4 tiers if you count the bottom, and each is about 6 inches deep. It has a nicely-fitting lid. It stands on fairly tall legs, which makes it comfortable to access and also provides good airflow for the bin (the tops of the legs are perforated).
The idea behind a tiered bin: as food and bedding are eaten and turned into worm castings, the worms will move up higher to get new food, leaving finished castings behind. In reality, this isn't quite how it works out- worms continue to eat their castings several times over, and it can be tricky to get all worms to move up in a bin. Still, the shallow tiers make it easy to maintain a healthy environment, and the round shape makes it easy to harvest castings.
There are also square tiered bins, though square bins can be slightly harder to harvest. Worms love to squeak themselves tightly into corners and in bunches, and the corners can make it somewhat harder to get all the worms out at harvest time.
A single-layered bin works fine, too. A 30- to 40-gallon opaque plastic storage tote or an old recycling bin can be used, provided the bottom and lid are drilled with lots of holes. The bin needs to be slightly elevated, say, on bricks, to provide good aeration, and there should be a tray under the bin to catch moisture. If the bin doesn't happen to have a lid, a well-perforated piece of black plastic will do in a pinch. A lid is preferred, though- sometimes worms go on the march, and a lid keeps them nicely tucked in.
Don't use cardboard. The worms will eat it and it won't hold up to moisture.
Don't use particle board. Who knows what mystery glue was used to hold the sawdust together- it could poison your worms and your castings.
Don't use metal. It is harder to maintain temperature and it can rust or deteriorate from the moisture.
Wood works ok, particluarly for outdoor bins. Like any other bin, it also needs to be well-aerated. Avoid wood finishes and do not use pressure-treated wood. It can be harder to maintain healthy conditions in a bin if a wood bin is kept indoors.
Folks, it pains me to say it, but for indoor bins, plastic works really well in this case. At least the Can 'O Worms is made from recycled plastic.
Any bin you use must be opaque and always kept dark. Worms don't like any level of light. This makes it fairly easy to harvest castings later, though.
Worms need a lot of airflow to be healthy. It's best to keep a bin out in a room, rather than tucked inside a cabinet or small closet.