Why do we allow users to edit others' posts?
Edits give the site a wiki feel, and allow the information contained on the site to constantly evolve and remain up to date. Importantly, if you see something that is misleading like the edit in question, which is a minor mistake, or which could use some additional links, images, or a schematic, please make the post better! Note that this doesn't include changes involving a major change in the meaning of a post, changes based on a difference of opinion, or changes to a question which affect the usability of subsequent answers.
Sometimes, the content is fine, but the post should be edited to better conform to the site guidelines. This could involve changing tags, removing unnecessary meta information from the title, removing signatures or salutations, and the like.
Additionally, and probably most commonly, edits ensure that the appearance of the site is top-notch. We'd like to be a professional quality community. Spelling errors, grammar errors, bad or nonexistent formatting, and unprofessional writing (txt speak, l33t speak, i dont like capitals or punctuation, etc) detract from that goal. This is actually the most common cause for an edit!
The privileges page reads:
When should I edit posts?
Any time you feel you can make the post better, and are inclined to do so. Editing is encouraged!
Some common reasons to edit are:
- to fix grammatical or spelling mistakes
- to clarify the meaning of a post without changing it
- correct minor mistakes or add addendums / updates as the post ages
- add related resources or hyperlinks
Try to make the post substantively better when you edit, not just change a single character. Tiny, trivial edits are discouraged.
Note that major changes to the original authors' intent is not among the common reasons to edit. Also, there's some mixed opinions behind the "tiny, trivial edits are discouraged" line. I've been known to edit a single character for a misspelling in a title, and we don't want to leave broken windows around, but generally, try to keep useful information on the front page from being drowned out by edits. While the post is on the front page, it's fair game for any edit which improves it. (Just try to do a thorough job, so that we don't convert posts to Community Wiki accidentally.)
To see this in action, you can take a look at my edit history, or Grace Note (another moderator) provides a good example to follow.
Who can edit a question or answer?
The original author of a question or answer may edit their own post. Users with a certain level of reputation (currently 2000) may edit any question or answer. The one exception is locked questions, which may not be edited by anyone, including the original author, until they are unlocked. Additionally, the amount of reputation needed to edit community wiki editable posts is much lower (currently 100) than that needed to edit ordinary questions and answers. If a user does not have enough reputation to edit directly, they can still suggest an edit (see a related FAQ question, How do suggested edits work?).
How can you tell what has been changed between edits?
Edit indicator
Once your question has been edited, there will be a note of it, with the time since the last edit hyperlinked to a revision history for the post:

Revision history
Each revision is displayed in a separate, collapsible section. Older revisions start out already collapsed. If a comment was specified by the person editing, that will be displayed in yellow next to the revision number; otherwise, the total number of characters added or removed in that revision will be listed. Once expanded, the revision will be displayed, with changes highlighted:

Revision diff color key:
- Green background: characters added
- Red text +
strikeout: characters removed
This answer borrows heavily from the meta.stackoverflow.com FAQ on edits.
Additional references at the FAQ entry on editing and the edit privileges page