What kind of backing should I look for to lay on tile floors?

Purists demand jute. But that's not best for tile.
The latex foam kind with ridges or waffle pattern won't slip, and it's not going to absorb the moisture that often goes with tile flooring in warmer climates.

Latex also might stick on smooth shiny surfaces if undisturbed long enough, however with tile you have to consider the non-slip characteristics.

With unbacked or Jute backed.. you can easily slip and fall even as an adult. With young kids around the latex backed is mandatory. The smaller the rub the more important it is.

If you have a larger area rug say 6"9 and larger it becomes less important... though a slip near the edges always remains a possibility.