A cross-examination is the questioning in court of a witness produced by the other side. The ability to test the evidence of one's accusers is a fundamental part of natural justice. For a recent explanation of the importance of cross-examination to the legal system see the statement by Lord Phillip in the United Kingdom's House of Lords:
The best way of producing a fair trial is to ensure that a party to it has the fullest information of both the allegations that are made against him and the evidence relied upon in support of those allegations. Where the evidence is documentary, he should have access to the documents. Where the evidence consists of oral testimony, then he should be entitled to cross-examine the witnesses who give that testimony, whose identities should be disclosed. Both our criminal and our civil procedures set out to achieve these aims. 
In a cross examination, the attorney will typically attempt to discredit points made by the witness during their direct examination by presenting them with leading questions that require a yes or no answer. A lawyer may not advance any theory of a case in court if they have not first given a witness the opportunity to deny such a theory.