It turns out there was a guy named Abraham Trembley back around 1740 who performed all kinds of experiments on the little guys to try to figure out if they were plants or animals.
Among other things, he discovered that they can move around using several different means of locomotion, that they can reproduce either sexually with sperm and eggs or asexually by budding, and most amazing of all, they exhibit regeneration of lost parts.
If you cut the head off of a hydra, it can grow a new head. If you cut a section out of the middle, the head end grows a new head, and the foot end grows a foot. Here’s where it gets interesting: if you graft together two sections from the middle of one or more hydras, and deliberately try to foul things up by grafting the two head ends together, the bigger piece wins and grows a head, and if both are exactly the same size, whichever piece was originally closer to its donor’s head wins. It gets even better – if you squish a hydra through a fine mesh, like a piece of silk, its components get scattered and it can’t regenerate, but if you heap them up into a pile, it can actually reorganize itself into a hydra.
Here’s a great article with historical background and some good pictures: Andrew Skolnick – Science and Medical Journalist – Smithsonian Hydra Article