There are actually two issues, the file type, which we call the container (things like .mov, .avi, etc) and the codec (the way the pictures are represented within the container).
The file type is the straightforward bit to advise on – FCP, being a native Apple product, works best with Quicktimes, i.e., files ending .mov
Codec is slightly more tricky. For quality keep the codec the same as it was edited with (you get some quality loss every time you process between different codecs). The disadvantages are size (editing codecs make large file sizes) and those files might not play on other computers because those other computers might not have the codec installed.
Web sites are harder to advise on as there are a handful of different files in use, so you need to speak to the website authors to see what they need. If in doubt, I have found that making a high res quicktime in H264 (picture) codec with the quality slider pushed up to the Best end and with .aac audio will usually be acceptable as that can be played by almost all computers and converted to whatever format a web designer needs without too much degradation.