Both stalagmites and stalactites are mineral deposits found in many caves that are rich in limestone. They form when water drips from the ceiling of the cave, leaving some mineral content on the low spot on the ceiling, and leaving some more mineral content on the place where it lands. These deposits, over thousands or millions of years, tend to grow as sharp points that pinch off the intervening space.
A stalactite is the shape which forms hanging from the ceiling, focusing the dripping water into a consistent downward point. A stalagmite is the shape which forms on the cave floor, just below a stalactite, and is often just as large.
There are several mnemonics which help people remember which is which.