6 of 8 edited to reflect YCor's comment

Any user with at least 15 reputation points should flag the post as spam. After 6 spam flags, the post is deleted. In addition to being deleted, 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 after a total of 6 flags that are either "spam", or "rude or abusive".

A detailed network-wide FAQ about spam can be found on Meta Stack Exchange: What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? That link, together with some additional links, can be found in the spam tag-info on this meta.

Some additional remarks:

  • Spam flags should be only used for posts that really contain spam. (A post is considered spam only if it advertises a product, service, or similar and is either or both unsolicited or lacks disclosure that the user is affiliated with the product/company/linked site/etc. This is elaborated in more detail in the network-wide FAQ.)
  • A spam flag automatically adds a downvote, so it's 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. Voting to close, when done in addition to flagging as spam doesn't cause any harm, other than spending one of your close-votes.
  • Editing the spam content out can be counterproductive - since other users who see the same post might not realize the it is spam. It's also possible that the spam post will be seen as a real post and end up giving the spammer significant reputation. On one Stack Exchange site, there's even been one edited spam question from a repeat spammer that got answers and made it onto the Hot Network Questions list. The spammer getting reputation makes it significantly harder to deal with their subsequent spam, as many of the automated checks for spam are bypassed for users with even a moderate amount of reputation.

So, basically, it can be said that once you've 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 reasonable - 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. If it doesn't look like spam just by glancing at it, you can use a custom moderator flag to explain to the moderators why it's spam.

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 could link to this post in your comment.
  • Theoretically, you can even mention the post in chat to attract the attention of other users.
  • You are always welcome to report spam, or posts you think might be spam, in Charcoal HQ. Charcoal is a project with the specific goal to help with detecting spam in the Stack Exchange Network. Charcoal has a chat room, Charcoal HQ, in which a bot, SmokeDetector, reports all the spam it detects on all sites in the Stack Exchange Network.
    The people in Charcoal HQ have a lot of experience with combating spam and can give an opinion on posts that might be borderline, including checking to see if any domain, phone number, or email has been used in previously detected spam. Mentioning in Charcoal HQ spam posts which you find also helps the SmokeDetector bot detect similar spam in the future, as members of the project will update the things the bot looks for to detect spam, so the spam you report is detected in the future. If users in Charcoal HQ think the post is spam, they're sure to add some spam flags to help the post be deleted faster.