MT on Twitter means "modified tweet," a form of copying someone else's tweet without using the automatic retweet button. Usually, it is followed by the username of the original author and a portion of their tweet.
MT can be contrasted to "RT" which stands for "retweet." Before the introduction of the automatic retweet button in 2009, users had to manually reproduce a tweet they wanted to share by copying it into their own Twitter feeds and crediting the original author. Since this led to larger character counts, the original text would sometimes have to be modified for length, leading to the creation of "MT" as a shorthand to indicate partial quotation.