1) To state that a certain fact is true. 2) In a civil case, to state that a plaintiff's allegation is true. If a defendant admits an allegation, the plaintiff need not prove it at trial. 3) In a criminal case, to state that a certain fact is true or to acknowledge guilt. 4) At trial, to allow, inter alia, testamentary or documentary evidence to come in for a judge or jury to consider in reaching a decision.
See, e.g. Ohler v. U.S., 529 U.S. 753 (2000) (admitting evidence) and Burley v. German-American Bank, 111 U.S. 216 (1884) (admitting an allegation).