Any user with at least 15 reputation points can flag posts. After 6 flags as spam the post is deleted. (And, in addition to that, even for users who can see the deleted posts, the content of the post is hidden - although they can still access the revision history.) The same thing happens if you flag a post as "rude or abusive".
A detailed network-wide FAQ can be found here: What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? (This link, together with some additional links, is also in the spam tag-info on this meta.)
Some additional remarks:
- It should be only used for posts that really contain spam.
- A spam flag automatically adds a downvote, so it is not necessary to downvote the post in addition to that.
- Also voting to close (or to delete) is not really necessary - it is likely that 6 users with 15 reputation points get to the post quicker than 5 users who have enough reputation to cast close votes. Of course, this assumes that the users know how to handle spam posts.
- Editing the spam content away might be contraproductive - since other users who see the same post might not realize the it is spam.
So basically it can be said that once you flagged the post, there is not much more to be done on your side. One possible exception is in cases when it might be unclear that the post is actually spam. (For example, if the post otherwise looks reasonably - perhaps because the content was copied from somewhere - but a punctuation mark or one word contains spam link.) In such cases it might be useful to comment - in order to make spam more obvious to other users. Or maybe even flag for moderators' attention.
Possibly - if you think that the spam might not be deleted quickly enough - you might leave a comment explaining to other users who see the same post what they should do. (For example, you can link to this post in your comment.) Theoretically you can even mention the post in chat to attract attention of other users - I will mention Charcoal which is a chat room related to and SmokeDetector which is a project with a specific goal to help with detecting spam in the Stack Exchange network.