Bistable & Monostable
Bistable means "two stable states", energised and not energised. It will be stable in either state, the input signal swapping its state between the two.
I guess a normal light switch is a simple example of a bistable device.
Monostable means "one stable state", a signal to it will cause its output to change state for a period of time and then revert back to its stable state awaiting the next input 'pulse'.
A 'push-to-test' button will do as an example - its output only maintains its 'unstable' state while the input (your finger) is applied; a monostable usually has a timing function such that its output will have a minimum 'set' time before reverting to its stable state.
Proper bistable relays can be bought, they have mechanical latching and a small solenoid coil to unlatch them. The two coils (set and reset) have seperate terminals to allow individual control of the two coils. They used to be used in telephone exchange systems (aah the good old days!).
I have never seen a circuit with a single pushbutton to 'toggle' a latching relay between states, though it must be possible - you basically need to construct a 'D-Type Flip-Flop' in relay logic, this could be complicated, maybe using five or more relays. It could be simpler to use electronics (a D-type flip-flop is a single IC).
Another idea (too stupid ?) is to use a latching pushbutton as used on some anglepoise lamps?, first press switches the lamp on, another press switches it off ???