The Jira REST API is a remote API. It responds to calls from other systems. It doesn't care what you use to talk to it, all it needs is an understandable request. Curl forms good requests, as do all other languages with a REST library. C#, Pro-IV, Java, Piaget, Pascal, Shell scripts etc etc etc - all are capable of talking to Jira's REST functions.
There is one thing that might be confusing - people often talk about the "Jira API" without saying that there are two. They usually mean the REST API for talking to it remotely, but there is also the internal API which is a Java API for writing code inside Jira. That's almost always for Server add-ons, it's totally different to the REST API, and, unless you're writing a Server add-on, not worth looking at.
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