Although this may add to your confusion, here goes!
RCI stands for Royal Caribbean International (its official name), as Ernie mentioned. I believe it also stands for Royal Celebrity International, since Royal Caribbean owns Celebrity.
RCCL stands for Royal Caribbean Cruise Line, and is the name many of we cruisers call it. By the way, Royal Caribbean uses this designation because, in addition to the website Donna mentioned, you can use www.rccl.com, and arrive at the official website.
Now the question is: what will they/we call it when Royal Caribbean merges with Princess..................?
The proposed merger is on hold pending review by US and European authorities to determine the effect on competition. Carnival's attempted hostile takeover is also subject to this review. Latest info seems to indicate that decisions will not be made and merger, if any, will not take effect until next Fall.