Infdev is a mode still under development, in which the playing area is said to be 8 times larger than the surface of the Earth. Currently, it generates strange landforms, including floating islands and strange looking cliffs that do not form normally in InDev. If you die, you can respawn at your starting point, but you drop all of your items you were carrying and can pick them up if you go back to where your body was (before the items expire).
Mobs and more realistic trees have been added on April 15th 2010. Few more key features from indev haven't yet been added, although Notch intends to re-add everything in upcoming updates.
Sometime in the future, indev will become Infdev, and indev save files (mclevel) may or may not be supported, Notch mentioned he will try to support backwards compatibility for indev but no promises are made.
You must buy the game in order to play InfDev.