Yes, they work without cables. What is confusing you is the fact that there are a number of partially overlapping definitions.
A range extender is something that works without cable. It picks up the base station's wifi signal, and re-transmits it suitably amplified.
Sometimes, a genuine range-extender will also have an ethernet dock. This is an extra feature that allows you to connect to your LAN a component without wireless capabilities. In this case, the range extender is moonlighting as a "wireless bridge". This is one of the confusions mentioned above. Manufacturers occasionally sell products which are capable simultaneously of both functions, at a premium of course.
You should also add to this the fact that having cable access to the range-extender makes configuration somewhat easier and more secure (no need to leave those open wifi networks around, while waiting for a configuration). So there is some incentive in providing an ethernet dock even for the pure "range-extenders", even though by no means all extenders have one such dock.
Keep in mind that whenever the connection with the base station occurs through a cable, the correct definition is "AP", or "wireless Access Point". Also, components which offer both wifi and ethernet connections, and connect to the base station via ethernet cable are generally routers, capable of providing more sophisticated services than an AP/wireless bridge/range extender. For instance, only a router can provide DHCP/dnsmasq/routing. Range extenders etc. have no routing capabilities, except perhaps for the purpose of configuration. On the other hand, I am aware of no component that provides both ethernet and wireless access, and does not provide router services.