Hashtags, at-signs, retweets, etc.

Twitter has given birth to some conventions used in "tweets", which have sometimes leaked into other social-networking and messaging as things posted to Twitter get re-posted to other services, or users accustomed to using and seeing these conventions start using them elsewhere. The result is some things which may seem mysterious to "data archeologists" going over archives of social feeds and chat logs from the early 2000s.

Hashtags
A hashtag is a sequence of letters and numbers preceded by the # sign (known variously as a hash, number sign, tic-tac-toe board, or, to Americans, "pound", which confounds British people used to their own currency symbol; INTERCAL calls it an "octothorpe"). The intent is to tag a tweet or message as pertaining to a particular topic to make it more easily searchable. Twitter automatically hyperlinks hashtags so that you can see other recent messages that used a particular tag, and "trending topics" are shown on the main screen of Twitter by some algorithm of theirs which finds tags that are being used a lot at the present time.

There is no official registry of hashtags; you can use any tag you want when composing a tweet. Sometimes the people in charge of an event, campaign, or other activity will come up with an "official" hashtag they try to get supporters, members, and followers to use, but they can sometimes get "hijacked" by people opposed to whatever is being promoted, so that when somebody looks up the hashtag they get largely snarky critical commentary.

Hashtags have showed up many other places besides Twitter; Google+ has made its own implementation of the concept, and they can be seen sometimes on other services even when no special functions are implemented for them.

Hashtags can name particular persons, places, or things: #Apple, #Thailand; or they can be cryptic abbreviations some in-crowd knows: #tcot, #ows; or they can be any other string somebody thinks makes sense to tag something with. A hashtag of #irony can be used ironically (or not-really-ironically, like the examples in the infamous Alanis Morissette song) if you wish.

At signs
The at sign (@) has a long history of usage in e-mail to separate the username/mailbox from the hostname/server/domain. However, in Twitter it has taken on a slightly different usage where it precedes the username, like @JaneSmith. This causes Twitter to hyperlink it to the user's public feed. Some other services also do a similar thing, but the usage can even be found in other places such as blog comment threads where there is no specific software function supporting it, but people feel the need to denote in some way that their message is in response or reference to another participant in the discussion, so they use an @ sign followed by the name or handle of the person they're referring to.

Retweets, etc.
A few all-capitals initialisms show up at the beginning of tweets to indicate particular things that have been done with them. RT stands for "retweet", meaning that the tweet was originally posted by somebody else but has been re-posted by a different user. It is followed by the at-signed username of the original poster. One might sometimes even see several RT @user sequences in a row to note a multiply retweeted message, but the 140-character limit of tweets results in these being trimmed pretty soon. Some variants are also used such as MT for "modified tweet", where somebody else's tweet is reposted in an altered form. Sometimes the alteration is just to save space, but sometimes it may be a snarky edit designed to make a tweet on a controversial issue say the opposite of what it originally did.