For more information, see "What are badges?" and the badges list in the Help Center.
Badges are awarded to users in recognition of their contributions to the community. There are many ways to contribute, and consequently, there are many badges.
There are three ranks of badges. Bronze badges are relatively easy to get, and often help teach users how to use the system. Silver badges are more difficult to earn, and can be gained for things like posting extremely insightful questions and answers, as well as a dedication to moderation and improvement of site content. Gold badges are the most difficult to earn, and generally signify outstanding dedication or achievement.
All of the badges that a user has earned are displayed on the user's profile. Also, each user's user card shows the number of badges the user has earned, broken down by rank.
Most badges a user can earn don't have any effect on site functionality; they are simply signs of accomplishment and bragging rights. A user's abilities are governed not by badges, but by their reputation points, with very little exception, other than the following two specific cases:
Holders of a gold tag badge (see below) can bindingly close questions with that tag as duplicates and reopen closed-as-duplicate questions that use that tag.
On Stack Overflow only, users need to attain four specific badges in order to be able to run in a moderator election, in addition to having a minimum amount of rep.
Additionally, certain specific badges can count toward the user's candidate score in a moderator election, which is a number shown to election voters that provides a rough indication as to the user's potential moderation skills. Aside from purely being shown to voters as info, this number isn't used anywhere else.
Users get badges by participating on the site. The badges page summarizes what is specifically needed to gain each badge. When a user meets the criteria for a badge, an automated background process adds the badge to the user's account.
Some badges can be earned more than once. If a user earns such a badge more than once, their name will appear on that badge's page once for each time they earned the badge. The badges listing on the user page will also display a multiplier next to the badge indicating the number of times it has been earned.
In most cases, badges gained or lost may not show up immediately, but will instead be awarded the next time the system recalculates badges, which occurs periodically.
These are awarded when you achieve a specific number of upvotes for answers within a particular eligible tag (which is in use on at least 100 questions). If you wish to track your tag badge progress you can do so, but only tags with 100 questions are visible.
There are three levels. Each level requires a minimum overall score sum on all answers you've posted to questions which use the tag, and a minimum number of answers you've posted to such questions:
Votes on questions and community wiki answers are ignored, and no tag badges are issued for tags that aren't used on at least 100 questions.
For example, if you get 400 upvotes for answers to questions tagged java, you'll get a java silver badge. Earn another 600 upvotes for a shiny gold badge, along with a binding close-as-duplicate vote on all questions tagged java.
You can also check your progress on the Stack Exchange Data Explorer.
See the announcement post and this section of the list of all badges and their exact criteria for details.
TL;DR: Regular (non-tag) badges you've earned will never be revoked except under exceptional circumstances. However, tag badges will be revoked if you don't qualify for them anymore.
Any tag badges you've earned will be revoked if you no longer have their required score or number of answers. This could happen through your answers being downvoted, deleted, converted to community wiki, or being dissociated from your account, or if votes on your answers are invalidated. It can also happen if the tag is removed from questions you've answered.
You'll also lose tag badges if they're no longer awarded (i.e., are for tags that are no longer eligible for tag badges as they're now used on fewer than 100 questions or are for tags that don't exist anymore altogether).
In most cases, regular badges, once actually awarded (see the last section), are yours to keep. Even if you no longer meet the criteria to earn the badge (the post was deleted, you got downvoted, the feature was removed, the badge criteria were redefined, etc.), you get to keep your badge. If you qualify for the same badge again (another post earns you a post-related badge, for example), you won't be penalized either (source).
The Stack Exchange team has stated repeatedly that non-tag badges will never be revoked except under the following exceptional circumstances (though not always):
For the last bullet above, behavior that qualifies as "heinous" is defined by Stack Exchange employees on a case-by-case basis, but here are some guidelines:
Under extremely rare circumstances, the team may deem it necessary to completely stop awarding one or more badges and retroactively revoke all prior awards of the badge(s). This has happened two times in Stack Exchange history so far:
Search to make sure it hasn't already been suggested, and then post it as a new question tagged feature-request badge-request.
However, keep in mind that badges are meant to reward positive behavior. Just requesting a new badge for the sake of having a new badge will likely result in your request being received very negatively. Make sure your badge request has meaning - explain exactly how the badge should work, the positive behavior it would encourage, and why that behavior isn't already encouraged by existing badges.
Most badges, except the Informed badge, aren't awarded immediately, but rather by a script that runs periodically. Be patient.
Note that if you cease to meet the badge criteria before the badge script runs and awards you the badge, you won't earn the badge. For example, if you ask a question with a positive score, but your question gets downvoted before the script runs, you won't earn the Student badge. (This does not apply if the downvote were to occur after the script has awarded you the badge; see Can badges be lost/revoked/taken away after they are awarded? above.)
If 24 hours have passed since you met the criteria, you still continue to meet the criteria, and you still haven't earned the badge, only then should you report it. See: How long does it take for badges to be awarded? How are they generated?
1. An example case where an "other mistake" happened was when the criteria for the publicity badges (Announcer, Booster, and Publicist) were changed to count questions and answers separately rather than count answer clicks toward the parent question. Users who earned a badge under the previous rules but wouldn't qualify under the new rules got to keep their badge, but those who earned one under both the old and new rules (and as such earned a duplicate badge) had their duplicate award revoked, retaining only one badge.
2. When an account is merged into another target account, only badges on the target account are retained as part of the merge process; badges earned by the source account will disappear. However, if the new, combined account qualifies for any new badges as a result of the merger, including badges originally earned on the source account, those will be awarded to the combined account when the relevant badge scripts run. As an example, if the source account continued to qualify for the Nice Question badge but the target account didn't, that will go away in the merge process, but will later be re-awarded to the combined account (since it now owns a question with 10+ score). This does mean that any badges that the source account qualified for or earned in the past, but the new, combined account wouldn't qualify for, will be gone until they're re-earned.
Example: the Nice Record made-up badge is awarded for asking at least 10 questions with at least 50% received positively. Account A asked 10 questions with 6 (60%) received positively and so earned and continued to qualify for Nice Record. Account B asked 20 questions with only 8 (40%) received positively, and didn't earn the badge. If both accounts are merged into each other, the combined account will have asked 30 questions, of which 14 (~47%) will have been received positively, and so wouldn't qualify for the Nice Record badge. If B is merged into A, leaving A as the new, combined account, badges earned by A will be retained, and Nice Record will be kept. However, if A is merged into B, then the badge will be gone since only those earned by B are kept, and as the new, combined account doesn't qualify for Nice Record, it won't be re-awarded.
Certain specific badges (e.g., Analytical and the retired Documentation badges) are awarded based on an internal user flag that gets set once an action is performed, which the badge scripts then check to award the badge. Upon accounts being merged, any internal flags set on the source account will be transferred to the target account. This means that if the source account qualified for such a badge, as before, it will be removed upon the merge, and then re-awarded to the target account (since it now has the flag set).