What is a chronometer? What is a chronograph?

These two terms are commonly confused among new watch aficionados. They actually have very little to do with each other.
A chronometer is a watch which has passed a test given by the Contrôle Officiel Suisse des Chronometres, or COSC. The COSC is an official Swiss government agency which tests watches to ensure that they fit within a narrow-but-usually-obtainable window of acceptable error (i.e., the rate in all positions falls into the range of -4 seconds/day to +6 seconds/day). While some watch companies tout their products as having a COSC certificate, it really is not that difficult to pass the test, and over 95% of the watches submitted pass. Another factor to consider is that the COSC does not test watches as they are sold in the store, but movements fitted with a temporary case, dial and hands. In addition, the COSC certificate cannot say anything about how the movement was handled after testing.

Several watch manufacturers actually put more stringent tests than the COSC procedures on all of their watch production (the JLC Master Control 1000-Hour series is probably the most well known.)

A chronograph is a watch that tells the time of day and also allows the user to time events of short-to-medium durations (i.e. from a few seconds to a few hours, typically). This is usually done in a mechanical watch through the central seconds hand, and one or more subdials (the regular, or continuous seconds, is also located on a dial). Chronographs are of varying usefulness, and are an interesting complication to put on a watch, as they often give the watch a sporty image. In addition to elapsed time, chronographs are often fitted with several scales designed to measure other things, such as pulse rate or units manufactured per hour.

A typical modern chronograph is operated with two pushers: one to start and stop the timing, and a second to reset the hands to zero when the timing is stopped. (In some older chronos with only one-button for control, the sequence of pushes was start-stop-reset - there was no provision to continue timing once the mechanism was stopped.)