How do white blood cells fight infection?

Organelles used.
The lysosomes are used so that white blood cells called macrophages can digest and disolve things they engulph.  I would also think (sort of an educated guess) the Rough ER, which produces proteins meant to be shipped outside the cell, is used a a lot because a lot of immunology depends on the immune cells ability to signal other cells, signals often coming in the form of expelled proteins.  The cell membrane is also important because chemicals serve as cellular "ID badges" and allow immune cells to identify which cells belong in the body and which don't.  These chemicals are usually made of glycoproteins (sugar molecules stuck to proteins) and are called "Antigens."

Eating them is simple way of explaining what they do to someone who has not had any biology training. The white cell surrounds and engulfs the bacteria cell then destroys it and the destroyed products are removed from the body in several ways. The one most familiar to you would be pus. They can also produce antibodies which are specifically designed to go after a specific organism and destroy them.