It is possible.
A SMALL subset of routers in their default firmware offer 'client mode' capability. This is what you describe, and is best.
A larger but still small subset of routers in their default firmware offer 'bridge mode' or 'wds brigde' capability where they can also bridge two physical wired ether net networks over their wireless link, one or more being the remotes bridged over to a main router that actually may have internet connectivity.
Sometimes there are limits in the types of encryption or authentication or other settings you can use in bridge / client mode. Sometimes the features only work between a certain model / manufacturer of client/bridge router and a certain (same or similar) model / manufacturer of linked router.
The BEST solution is to get a router which in its manufacturer, model number, and HARDWARE REVISION is compatible with THIRD party firmware like DD-WRT. This third party (often free) firmware is FAR better in supporting client mode / bridge mode and other advanced features than most routers are when they run their stock off the shelf firmware. Only a few dozen model / revision routers are compatible with the DD-WRT or TOMATO or other similar 3rd party firmwares, especially the best / full / mega version of these firmwares. So you must choose your model very carefully for best results.
http://www.dd-wrt.com/wiki/index.php/Supported_Devices
Furthermore some companies like linksys, netgear, dlink et. al. offer "wireless gaming adapters" which are intended to connect a wired ethernet based game unit like a PS3, XBOX, etc. over wireless to a main router and internet connection. These units could probably be used with general computers as well. They are intended to work "off the shelf" with only a little configuration, so if you do not want to replace the router firmware, maybe buying one of these may be easier for you.