"Hydrolysis" is using water molecules to split other molecules apart.
Many of the molecules our bodies work with are long molecules held together by covalent bonds (strong bonds where electrons are shared between atoms).
When such a bond is broken with hydrolysis, one part of a water (H2O) molecule - the hydrogen atom (H) - forms a covalent bond with one part of the targeted molecule and the other part of the water molecule - the hydroxyl group (OH) - forms a covalent bond with another part of the targeted molecule.
The result is that the target molecule is split into two separate molecules, one with an extra hydrogen atom (H) and one with an extra hydroxyl group (OH).
A generic example should help explain this:
In the example above, I underlined the water molecule [H2O] that was used for hydrolysis as well as the parts of that water molecule that ended up in the two newly formed molecules [H and OH]. Each dash [-] represents a covalent bond between segments of the molecule that are represented by a number in parenthesises [(1), (2), etc.].
In this example, the original five segmented molecule (on the left side of the equation) is split into a three segmented molecule and a two segmented molecule (on the right side of the equation). (1)
As we discus on ATP & cell metabolism enzyme-catalyzed hydrolysis is used to remove the phosphate group from ATP so that it can be transferred to whatever molecule the cell wants to make less stable (more reactive). (1)
1This explanation of hydrolysis as well as the example provided is based on information from the textbook
Biology, sixth edition by Campbell and Reece, Copyright 2002 by Pearson Education, Inc.
Hydrolysis is discussed on page 63 of Chapter 5, "The Structure and Function of Macromolecules"