A “contactor” is essentially a coil of wire that, when energized, creates a magnetic field that causes its contacts to move. This can cause two contacts to touch each other, thus completing a circuit, or can cause two contacts that had been touching each other to move away from each other, thus opening a circuit. It can also be set up to do both (i.e., open one set of contacts and close another set of contacts). There will be one or more springs that would pull the contacts in one direction, and the magnetic field from the coil will pull them in the other direction. Thus, with the coil deenergized, the contacts are in one configuration, and with the coil energized they move to the opposite configuration.
You can set it up so that a “main” set of contacts does one job (such as turning on a motor), while at the same time one or more other sets of contacts (“auxiliary contacts”) are doing other jobs (e.g., turning on a light to tell you that the motor is running). You can use a single contact, such as from a pressure switch or a level switch, to energize a coil of a contactor, and then have several sets of contacts from that contactor to do a number of jobs. That is what is meant by “contact multiplication.”