In theory, a POST request should attempt to create or modify some resource on the server. As @Tichodroma pointed out, an idempotent request will affect this change only the first time it is sent, but otherwise what's important is that some state on the server will be changed by the request.
More practically. POST requests are often used when the request payload is too large to fit into a GET URI (e.g. a large file upload). This is usually an intentional breach of HTTP standards to avoid a 414 Request-URI Too Long
response.
In terms of verbiage, I don't know if I like "process", because even a GET request will usually be "processed" to determine the resource to return. The main difference in my mind is the change of some state on the server.