Sometimes it gets auto-deleted (I think if it receives enough of certain kinds of flags, or if it contains certain words - this mostly relates to getting rid of rude/abusive comments), but normally it just ends up in the moderators' dashboard for us to review.
Unless it's obvious from context that the comment stands alone and should be deleted, I normally visit the post and review the whole thread to check for related comments that become invalid. But it's easier if you do flag all the comments that need attention, not just one.
If I think a comment should be deleted, I'll also check that the flag matches the reason I think it should be deleted, and if it doesn't I might decline the flag first then delete the comment anyway, particularly for the more negative ones like "unfriendly or unkind" or "rude or abusive".