Discovery is a pre-trial phase of a lawsuit. It's a process whereby the other side's lawyer will ask a series of quesitons to "discover" a story.
There are two styles of discovery. One style proceeds simply as an exercise to understand the plaintiff's claim. Luckily for the plaintiff, this is the most common style of discovery. The second style is conducted more as a cross-examiniation. The lawyer asking questions tries to lead the person answering questions into answers that harm his or her case.
Cases often settle before the discovery phase, so not all clients have to go through this process.
Read what you can do to prepare for a discovery here.