Think of the normal meaning of the word. The residue is what's left over after whatever was going to happen has finished happening.

For amino acids, the interesting thing is that they get joined into chains that fold up into proteins (which do all the work). The residues after that happens looks like this:

            R   O               R   O
        H   |   ||          H   |   ||            H
  ----- N - C - C --------- N - C - C ----------- N etc
            H                   H

The lines - and | are chemical bonds. The Oxygens are double bonded. N= Nitrogen, C=Carbon, O=Oxygen. The R is one of 20 different chemical groups called side chains that make each amino acid different.

When the amino acids are isolated (not chained) the double-bonded carbon on the right carries an extra OH group. The whole right-hand carbon is often written COOH. The nitrogen on the left hand side of each amino acid carries an extra H. As part of the process that joins the amino acids together, the OH on the left and the H on the right pair off to form water. This is called condensation, and it happens at every junction between two acids. So if there are n amino acids, there are n-1 junctions and n-1 water molecules that were present (in aggregate) in the constituent amino acids that don't make it into the final protein chain.

Note, though, that one of the methods of chopping up protein chains is adding the water molecules back again, so you should know exactly what you're looking if you want to count masses precisely.

Jeez. Didn't mean to write all that. Hope it helps.