The size of the tank depends on how large a group you want to keep. Normally you should keep 6-8 animals since they are schooling fish. They don't have to be the same species, you could have 2 each of 3 species.
There are two exceptions:
1: Dwarf species should be kept in groups of about 10 (or more). Some of them swim in mid water and might be shy if they are too few.
2: Many long nosed species are VERY territorial (not the cute lite fish we associate with Corydoras).
To keep a group of 6-8 animals follow these suggestions.
The really large species like C. semiaquilus, narcissus, B. multiradiatus and britski need a tank of about 200 l (50 gal). Even that might be too small since some of the large Corydoras and some of the long nosed are VERY territorial and will kill the other fish of the same species!
Large species like C. barbatus need a tank of more than a 100 l (25 gal). Long-snouted males can be very territorial and the largest male might kill the smaller ones! I only had a male C. cervinus left after he has killed 3 females. Even the fry can be territorial. Who said that all Corydoras are nice and friendly?
The slightly smaller like geryi, viginiae and C05 between 75 and 100 l (20-25 gal).
The bulk of the species will do nicely in 30-50 l (8-13 gal).
The dwarf species like hastatus, habrosus, and pygmaeus can be housed in 20 l (5 gal).
In the same tank you can of course keep other fish. Remember never to overstock though!
Tank sizes for breeding purposes see Breeding.

Filtration

Some of the more sensitive species (i.e. nijsseni and concolor) need a very good filtration. Otherwise they are not that fuzzy.
For small tanks less than 50 l (13 gal) a LARGE sponge filter (larger than fist-size) with good air flow can be used. Remember to clean it regularly (at least once a month).
For larger tanks an internal power filter or an external canister filter is among the best.
Do like this to keep the good bacteria culture in the filter media when you clean it:

Fill two buckets about half ways with water FROM THE TANK. (NEVER use tap water since the chlorine will kill off parts of the bacteria.)
Take out the sponge/filter media and squeeze it repeatedly in one bucket.
Move it to the other bucket and squeeze repeatedly again
Put it back into the tank/filter canister.
Fill up the tank.

No matter how large your filter is or what type of filter you use you NEED to change water! See Care for more discussions about water.

Water

I have a lot to write about water, RO water and copper in the tap water, but will get back to that later on.

Lighting

Not critical but if you keep plants in the tank you should adapt the light level for the plants. Some species are easily scared if the tank is to well lit and will lurk in the shadows. On the other hand, a too dark tank in a well lit room can also be bad since you will cast a shaddow on the tank and therefore scare the fish. A species that huddles under some roots almost never to be seen in one tank, will be laying out in the open at the front glass in an other. I just don't understand why, maybe the lighting and room background?

Substrate

Whether to use bottom material or not can be debated. The most common is to have some kind of gravel or sand. When a bottom material is used it will help increase the surface for good bacteria to multiply.
A problem with Corydoras is eroding barbels. Some people say that having sharp gravel can erode the barbels on Corydoras. I DON'T believe in that. Of course glass or something similarly sharp would be bad!
I have identical gravel and sand in all my tank. Some have pie size gravel and the other have the finest beach sand you can find. In both these tanks I can see barbel erosion on the more sensitive species if the water quality is bad (meaning I don't change enough water and the filter is not cleaned often enough). Some of the more sensitive species are C. concolor, habrosus and wild caught paleatus (at least those from Uruguay).
If you look at a healthy Corydoras working it's way through the gravel you see that the barbels are not sensitive to wear. They move rather large gravel without a problem. The problem is probably bacteria build-up in the gravel if you don't do your maintenance.

Advantages with bottom material:

Some species prefer a dark bottom, other a pale one. Some pale Corydoras show a better contrast on a pale bottom. You could also paint the bottom.

Many (all?) species "like" to probe around in the bottom for food. The long nosed species really seem to be built to do that.

Less risk of fungus attack on fry.

No reflections from the bottom (paiting the outside black will reduce the reflection).

Disadvantages with bottom material:

Difficult to see if all food has been eaten

Difficult to clean without vacuuming out sand

Heating

Keep the temperature at 22-25°C (72-78°F). If you need a heater depends on the ambient temperature and the heat generated from the lighting. Some species can be kept at higher temperatures (i.e. C. sterbai, concolor, and julii) 28-29°C (82-84°F), while some of the highland species (i.e. C. barbatus and macropterus) must usually be held at temperatures lower than 22°C (72°F) to survive for a longer period in the tank.
I've had a pair of C. barbatus that has survived the Swedish summer of 2002 without any problem. It was rarely below 26°C (79°F)! Keep them in the lower tanks if you have several to choose from.
The C. paleatus I've collected in Uruguay in 1997 and some C. undulatus I've bred survive nicelly in my largest tank with 28-29°C (82-84°F) even though they come from the cooler parts of South America!