The titular question came up in another thread.
This answer is out of date. The current flag procedures are described in another answer.
From the meta.so faq on 10k tools:
Users with sufficiently high reputation (10k on mature sites, 2k on beta sites) can see posts (questions or answers) flagged as offensive or spam.
♦ moderators can see those, as well as posts flagged for moderator attention.
From the meta.so faq on comment voting and flagging:
Comment flags are visible to moderators, but are also handled automatically - if enough users flag a comment, it is automatically deleted.
The accepted answer is out of date. The current flag procedures are described here.
Visible only to moderators
- other (needs ♦ moderator attention)
- all comment flags
Remark. Flags of types 1, 2, 4 can be handled automatically by the system, if enough of them are raised. The post or comment gets deleted in this case, and the flags are marked helpful.
Visible to moderators and to users with access to Low Quality Review Queue
- very low quality
- not an answer
Remark. Most of these flags are resolved in Review Queue, one way or the other. They only enter the moderator queue if one of the following happens
- The flag remains unresolved for an hour.
- The outcome of review is to recommend deletion, but the answer has a positive score or is accepted.
- The outcome of review is mixed.
The rules are documented here.
Visible only to users with access to Close Vote Review Queue
- duplicate question
- question should be closed for another reason
Remark. These are sometimes called closeflags; their classification parallels the classification of closevotes. Users below 3000 rep cast closeflags; users with 3000 and above cast closevotes.