The most famous aspect of the Songkran celebrations is the throwing of water. The custom originates from spring cleaning aspect of Songkran. Part of the ritual was the cleaning of images of Buddha. Using the 'blessed' water that cleaned the images to soak other people is seen as a way of paying respect and bring good fortune.
It also doesn't hurt that April is the hottest part of the year in Thailand, so being soaked is a refreshing escape from the heat and humidity.
Nowadays Thais will walk the streets having 'water fights' using containers of water or water guns, or stand at the side of roads with a hose and soak any one who passes by.
You may also get covered in chalk, a custom originating from the chalk used by monks to mark blessings. This combination of water and powder is almost identical to the celebrations of Holi, and indeed, it maybe that the customs originated in India as certainly Songkran is celebrated more widely and longer in the Northern part of Thailand.
Even more interesting is that the custom of water fights pops up in a distant place at the same time of year. Dyngus Day is celebrated on Easter Monday in Poland, and on the Monday the custom is that boys get to soak girls with water, but on the Tuesday, the girls get to throw crockery at the boys - again the water and powder combination.
As mentioned, Songkran is a Spring Cleaning Day, both physically and spiritually. On the physical side, in addition to cleaning other customs are that anything old and useless must be thrown away or else it will bring bad luck to the owner. On the spiritual side, some people make New Year resolutions.