Routing is a term with multiple meanings in different disciplines. In general, it refers to determining a path for something. In telecom, a call may be routed based on the number being dialed or some other identifier. In either case, a path is determined for the call.
Mail is also routed—I’m not talking about email here (though email is routed, too)—but rather, snail mail. When you write an address and a zip code on a letter, you are providing the means for the post office to route your letter. You provide a destination and, usually, a source address, and the post office determines the best path for the letter. If there is a problem with the delivery of your letter, the return address is used to route it back to you. The exact path the letter takes to get from its source to its destination doesn’t really matter; all you care about is that it (hopefully) makes it in a timely fashion, and in one piece.
In the IP world, packets or frames are forwarded on a local network by switches, hubs, or bridges. If the address of the destination is not on the local network, the packet must be forwarded to a gateway. The gateway is responsible for determining how to get the packet to where it needs to be. RFC 791, titled INTERNET PROTOCOL, defines a gateway thusly:
Gateways implement Internet protocol to forward datagrams between networks. Gateways also implement the Gateway to Gateway Protocol (GGP)  to coordinate routing and other Internet control information. ...