Twitter's a service, not a software that you can install. Basically, it lets you send "tweets", short messages of 140 characters or less, which other people can then follow. You can also follow other people's tweets, if you want. There are a lot of big businesses and big name celebrities on Twitter.
About the term micro-blogging, it simply refers to a form of blogging where you simply send short messages instead of writing full page articles.
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