What does the compartor do in Minecraft?

Answer 1

A comparator compares two signal strengths and outputs the difference.

In minecraft redstone signals work from a 15 block length:

When you place a torch with some redstone dust next to it, this redstone dust is 15, if you place redstone dust next to this redstone dust- the new dust is 14 etc. Once the signal equals 0 you cannot use the signal anymore.

If you place a comparator between two redstone lines (with one line in the back (flat end of the arrow on the top of the comparator) and one/two lines in the side, then any dust out of the front (pointed arrow) will be the remainder.