The X-Men are a paramilitary team of mutants who work covertly to advance the cause of mutant and metahuman rights. Most of them are graduates of Xavier Institute for Higher Learning, a school for 'gifted' peoples. Students may be recruited from Xavier's, but in general students are completely unaware of their activities or even existence, and are discouraged from joining until they are old enough and experienced enough to safely participate.
X-Men describes the group at large and is used when referring to their activities, sanctioned or not. Their actions are often covert, vigilante, and sometimes illegal (if just), so the X-teams maintain a high degree of secrecy. Some X-Men operate independently, usually in deep cover or as agents gathering intelligence, and may even be unknown to other X-Men, as they report directly to Professor Xavier or Jean Grey.