A Twitter bug is an issue, or an error with the site or a Twitter app.
Twitter bugs happen. Sometimes things go kablooey on Twitter; maybe the server gets overloaded with requests or maybe it returns something you weren’t expecting to see. You can report bugs here or send a friendly Tweet to @support; that is, the Twitter support desk to let them know what you’re seeing. Send a screenshot if you can.
The most famous Twitter bug, which is really an "over capacity warning", is the Fail Whale. When Twitter is over capacity, you see a cheery image of a whale being carried by little birds. There's no sense in reporting this to @support, because you won't be able to Tweet anyway.
Back in the day, you might see the Fail Whale for half the day, but these days it only shows up for moments at a time. Sometimes it's up for a second, only to be gone when you refresh the page. Other times, it's up for a minute or so. The only time you really see this bug / whale now is during major world events. Even then, Twitter has been holding up during times like these for quite a while now.
Then, for a year or so, there were a lot of follower bugs. For example, you'd suddenly have no followers, or suddenly it looked like you had unfollowed a ton of people (and sometimes you had!). Then, there were the bugs that actually allowed you to control who followed you. I haven't seen or been a victim to these follower/following crimes in some time, so I think they've got those sorted out for now.
Here are a couple examples of people sending Tweets to @support. You can see that half of the time they're just gripes, cause that's what you get for running a free service - users who think they own ya!
Why. Why do RT replies now have to include the @ of the person who RT'd it too...why @Twitter...why @support...why...
Hey @support, Y U suspend @anon2world? | #YAN
Hmmm. When I change the trend an error pop up on my iPhone. Why? @twitter @jack @twittermobile @support
Dear @Twitter @Support it's been 2 days since @FlyOverAngel account was erroneously suspended. Please right this wrong. #TwitterGulag
.@twitter @Support When do we get two-step authentication to stop GLOBAL PANICS from hacked accounts? cc @AP
What does a musician need to do to be included in Twitter Music? I'm on @Spotify and @Rdio already. @support?
As mentioned, there are quite a few gripes here. Lots of these requests are people begging to have their profiles verified, despite probably having very little need for it. Twitter, by the way, only verifies accounts with high rates of impersonation.
The other mostly just complain about features they like, don't like, or want. It's rare to see most of these requests even responded to, so you might be better off going straight to the source with your request over on support.twitter.com/forms/general.
The truth about these bugs is that you're going to wait a decently long time for a response. I once had an account suspended for six months, only to finally get the response that the account was "accidentally" suspended due to an algorithm update. Their "ticket" list is miles long, and they run a free service, so try to brace yourself if you ever run into trouble. The community on Twitter can also usually help you with common glitches.