Crampons where first used by Europeans in the early 1900's. These early crampons did not have front points and therefor required the user to cut steps on all but the gentlest slopes. They where however an improvement over the nailed boots which they eventually superseded.

Walking Crampons:
Walking crampons are generally graded C1 and have 10 points. This make them lighter than the other types of crampons but not as good on steep ground. They are normally made from hardened steel, which is very durable. Lighter and less durable Aluminium walking crampons are also available.

These crampons are quite flexible and will therefore attach to any crampon compatible boot normally using a classic strapped binding system, as shown below.

Walking crampons are ideal for glacial travel and general mountain walking but they will not give as much security or support as mountaineering crampons on steeper ground. They will perform well on snow but have shorter, stubbier front points compared to mountaineering crampons so they will not be as good on rock or ice.
Mountaineering Crampons:
Mountaineering crampons are normally graded C2 and have 12 points for added security on steeper ground. They are also normally made of hardened steel which provides the durability needed for the abuse they are likely to be given.

These crampons have a small degree of flex to allow comfortable walking and still offer a large degree of support. They are normally paired with B2 or B3 boots as B1 boots will not offer the rigidity required for typical mountaineering activities.
 
These crampons are attached to the boots using a number of different binding systems. The classic binding system (mentioned above) can be used but most users will opt for a more secure binding system like the new matic (also known as lever lock) system.

Mountaineering crampons are designed for all mountaineering activities up to about Scottish grade V ice. They are equally at home on classic Welsh or Scottish buttresses and gullies as they are on long alpine routes or Himalayan climbs. Although slightly heavier than walking crampons due to the extra and longer points mountaineering crampons are the standard winter crampons for most people.
Climbing Crampons:
Crampons designed more specifically for ice or mixed climbing are almost always graded C3. These crampons differ from the walking and mountaineering crampons because they are designed to offer maximum support when the user is standing on the front points only rather than for walking. The front points of climbing crampons will be vertically aligned to allow better penetration of ice and more precise movement on rock.

Climbing crampons where designed to be used with fully stiffened B3 boots. They can attach to the boots using the lever lock system mentioned above under mountaineering crampons. Some C3 crampons use a wire toe bail  which when combined with a heel lever lock provides a very secure attachment.

Climbing crampons are ideal for steep ice and mixed climbing whether it's on a valley ice crag or a technical mountain route.
Summary:

Crampons are available for most eventualities but a good pair of mountaineering crampons are probably the most versatile. The crampons must be more flexible than the boots they are used on, with this in mind the C rating system was designed to match the boot B system.

Obviously any crampon is better than no crampon in icy conditions and people have used walking crampons for climbing and climbing crampons for walking, they just wont work as well!