The "center of mass" of any object can be thought of as the average location of all of the mass in that object. For a simple object like a meter stick, the center of mass works out to be just the center of the object (i.e. the center of the stick at the 50cm marker). If you took a meter stick and put a big lump of clay at one end to make it heavier, then the center of mass would be closer to the heavier end.
A related concept is that of the "center of gravity". If you add up all the effects of gravity pulling on every part of an object, it appears to pull the center of gravity straight down. In almost all cases you are likely to encounter, the center of mass and the center of gravity are the same. (The only times this wont be true is if you are sty dying objects that are so large that the gravitational field due to the earth is not constant over the object, like for example the moon).
In order to balance something, all you need to do is make sure that the center of gravity of the object is either directly above or directly below the pivot point.
If the center of gravity is below the pivot, then the system is said to be "stable", which means that even if you give it a little push it will still be balanced. An example of this is holding one end of a meter stick gently between two fingers and letting the stick hang down below your hand like a pendulum. If you give the stick a little push, it will just swing back down until the center of mass is below the pivot point (your fingers).
If the center of gravity is above the pivot, then the system is said to be "unstable", which means that if you give it a little push it will tend to fall over. An example would be balancing the meter stick on the end of a finger with the stick pointing vertically up. If you do this you will find that the stick wants to fall over, and you need to keep moving your finger around to keep this from happening.
(published on 10/22/2007)