Geysir is without doubt the most famous geyser in the world.
Geysir is both the name of a geyser located in Haukadalur valley of Iceland, and of the geothermal area surrounding it. The geyser is sometimes distinguished from the surrounding area by calling it the Great Geysir. The Great Geysir is not the highest spouting geyser in the world (that would be Steamboat Geyser in Yellowstone), nor the most frequent, but it is the oldest known, having first appeared in 1294 CE after a series of earthquakes hit the area. It became quite famous in Europe, as few geysers were know at the time, and all of them were in Iceland.
Geysir gave its name to the phenomenon of geysers the world over (or within the Romance Languages, anyway). 'Geysir' comes from the Icelandic verb gjósa, meaning 'to erupt'.
The Great Geysir erupts irregularly and infrequently, sometimes laying dormant for years at a time, other times erupting every 30 minutes. Activity tends to peak after seismic activity, and then taper off. In the early 1800s the geyser was undergoing a period of dormancy; scientists decided to experiment and eventually found that they could stimulate an eruption by dumping soap into the geyser. (I have not been able to find a convincing explanation of the chemistry behind this, but apparently soap will cause most geysers worldwide to erupt, if used in significant quantities).
When the Great Geysir does erupt it can spout 60 meters high, although lately it has only been sending up spouts of 8 to 10 meters. It is, however, spouting more regularly than it has for most of this century. After earthquakes in 2000 it started spouting up to 8 times a day, but is currently (2008) down to once every couple weeks. As far as I am aware, they no longer use soap to stimulate eruptions.
If you get tired of waiting for the Great Geysir, you can head over to Strokkur, a smaller geyser in the Geysir area that erupts about every 8 minutes, sending boiling water up to 20 meters in the air. The Geysir area has 30 other geysers and hot pools, and numerous fumaroles and mud pots. There is also an impressive sinter. Even with these riches, Geysir is one of the smallest thermal areas in Iceland, at only 3 square kilometers. Small or not, the Geysir area is one of the more popular tourist attractions in Iceland.