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A pair of simple definitions would be:
Forward Proxy: Acting on behalf of a requestor (or service consumer)
Reverse Proxy: Acting on behalf of service/content producer.
Forward Proxy vs. Reverse Proxy (2012) explains the difference between forward and reverse proxies very clearly.
qyb2zm302's answer nicely details applications of proxies, but it slips up on the fundamental concept between forward and reverse proxies. For the reverse proxy, X → Y → Z, X knows about Y and not Z, rather than vice versa.
A proxy is simply a middleman for communication (requests + responses). Client <-> Proxy <-> Server
The proxy acts on behalf of the client. The client knows about all three machines involved in the chain. The server doesn't.
The proxy acts on behalf of the server. The client only knows about the proxy. The server knows the whole chain.
It seems to me that forward and reverse are simply confusing, perspective-dependent names for client and server proxy. I suggest abandoning the former for the latter, for explicit communication.
Of course, to further complicate the matter, not every machine is exclusively a client or a server. If there is an ambiguity in context, it's best to explicitly specify where the proxy lies, and the communications that it tunnels.