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.
As mentioned, Songkran is a Spring Cleaning Day 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.