Activities centered around observing whales (and occasionally other marine mammals such as dolphins) in their natural habitat. Generally recreational this can also be done for scientific or educational purposes.
In general whale watching consists of travelling to an area known for it's high concentration of whales (possibly due to abundant food or their migration routes crossing it the area). Often this is done via a boat or other marine craft although it is possible to spot whales from land in some areas.
Although typically recreational there are also scientific studies focused on observing whales to understand the current population numbers of different species and their patterns of movement.
Tourist focused "Whale watching" trips often encompass a larger collection of marine mammals, including dolphins and porpoises as well as whales.
These trips may be offered on a relatively local scale or entire holidays with intercontinental flights or cruises can be organized around the chance to spot whales and other marine life.
For full history and other details see Whale watching on Wikipedia.