Compression fittings are mainly used on copper pipe, they are very simple in function. An olive, a small ring of brass not unlike in appearance to a wedding ring, fits over the copper pipe. Then when the fitting around it is tightened they will deform the olive so it grips the pipe. Compression fittings are usually stronger than solder fittings by a small margin but are harder to airtight and don't look as nice. Compression fittings are also alot more expensive, some as much as 3 times more expensive in larger copper sizes.