A chronograph is a watch that has a stopwatch function. Typically, the top pusher will start and stop the chronograph. The bottom pusher will reset it. The chronograph generally can not reset in the running state. Special Flyback models allow the chrono to be reset at any time.
A chronometer is watch that has passed a series of tests, and is a superior timekeeper. Many watchmakers will put their movements through this test to illustrate their accuracy. The dial will typically have the word "Chronometer" or "Chronometre" on the dial.
To become a chronometer, the watch movement must pass 15 days of severe tests. The accuracy of the movement is checked in 5 different positions at varied temperatures. This simulates conditions under which the watch might be worn. The watch must average between +6 and -4 seconds per day in order to earn the certification.
If a watch is described simply as a "chrono", it is generally safe to assume it is a chronograph.