In a way a relay and a contactor are the same - both use an electromagnet to operate the switch contacts. The difference is more of size and application.
As a general rule "contactors" are high current, say 20A and above, and usually single-throw (i.e. on/off only, but can be 2, 3, 4, etc poles that are simultaneously switched).
Relays come in all sorts of shapes & sizes and are generally low-medium current, so some have gold contacts for "dry" switching of signals in the microvolt / milliamp range, but most have some sort of silver alloy contacts for up to the low tens of amps range. Often relays are available in double-throw (i.e. change-over contact) configurations with commonly 1 or 2, sometimes 3, rarely 4, simultaneous sets of contacts.
Also some relays are magnetically latched so have typically two coils and you momentarily energise one to set it "on" and the other for "off" (but with changeover contacts that is really positions 1 & 2, etc).