Discovery is the process by which lawyers engaged in a civil lawsuit make inquiries of opposing counsel, through formalized interrogatories. The goal is to discover as much relevant information as possible, to ensure that truth comes out at trial.
Texas has extremely liberal discovery rules, allowing lawyers to make almost any inquiry that is "relevant," with relevance defined very broadly.
Discovery is an expensive process. It can easily be abused to escalate costs on opposing attorneys, by asking pointless questions that still must be answered according to legal decorum. It can also be abused in "answers" - a common legal "trick" is to, in answer to a simple but probing question, enclose a highly important document in 2 boxes of meaningless documents. Such abuses can lead to sanctions, though (see Rule 37).
Even after a verdict has been rendered, discovery can be used to find assets to execute against to satisfy a judgment.