Each of these classes can do healing and have healing spells which can heal their current target. In addition, each healing class has a type of healing that they excel in. For example, Shamans are known as good multi-target healers while Paladins are known more for there ability to excel at healing single targets.
There are 2 main types of healing that a healer can perform, single target healing and multi-target healing.
Single target healing is usually performed on a tank or single target that is taking heavy damage while multiple target healing is used when more then one player is taking damage (in some raid encounters the dps classes may be taking damage and should be healed). In a raid situation healers are usually assigned a role of what type of healing they are to do.
Priests have a variety of different healing spells and can do each type of healing well. For example, a priest do can single target healing using the greater healing spells as well as use Circle of Healing (if speced for it) for multiple target healing.
Druids have multiple types of heals, giving them a fair amount of versatility. Most of the druid's healing comes in the form of a HoT, or a heal over time. This allows the druid to concentrate on one target (ie. the tank), and still be able to spot heal other players as needed. Depending on spec, a druid tends to either focus on big heals, like a priest, or on keeping the target constantly healed through HoTs, layering them as needed. Druids also have a nice emergency heal, in the form of Tranquility, which is a good-sized multi-target, channeled heal. Tranquility should be used sparingly, however, due to the threat it generates.
This section will contain information on paladin healing
This section will contain information on shaman healing