Yep, you said it. A bridge would typically be used where you need to link
two segments of the same network, with the same IP range, but which are
physically separated from each other. For example two buildings linked by
fibre-optic cable.
A router allows one network to talk to another, where the two networks are
intentionaly separate and use differing IP ranges.
There are in fact two distinct meanings of the word 'router.' Classically
it means the above. However, through common parlance it has come to also mean
a NAT router - the kind of box you use to connect your internal computers to
the Internet.
A NAT router is not quite the same thing as a standard router, The NAt
router's function is to allow a number of computers to masquerade as a single
IP address on the Internet. The two are often confused.
AFAIUI A repeater is simply a signal-amplifier for long cable runs. It does
nothing in the way of changing the signal's content.