When is the proper time to use each of these words?
Islam is the faith and the civilisations or communities built up around this faith. It literally means submission to Allah.
A Muslim is an adherent to this faith. This literally means one who submits to Allah.
Moslem is a variant on the word Muslim.
The main requisite to be a Muslim, one who is of the faith of Islam, is that he or she believe and submit to the one and only deity, Allah, for whom Mohammed was the final prophet. There are other threads currently in GQ which go into discussions of what this does and does not include.
BTW, I just reread my last post, and the final line may lead you to infer that this is my personal belief. Infer that at your own risk.
[sub]you could also infer from my UserName that I am of Judaic persuasion, once again at your own peril[/sub]