If you aren't familiar with the basics of Twitter, you're unlikely to be able to get very many petition signers from it. There are lots of sites that can give you a basic primer on using Twitter. One of our favorites is this page from Mashable, the social media news site.
A hashtag is a way of tagging a tweet as being related to a specific topic. It's simply any word preceded immediately by the number sign (#). You can find popular hashtags by searching for words that are relevant to your issue and seeing what hashtags are commonly used.
Including "@username" in your tweet is a good way to direct your tweet at specific people, even if they aren't following you. Direct your tweets at people who have a lot of followers and who you think might be interested. If they retweet you, that will be a big boost.
RT is Twitter shorthand for "retweet," which is when someone copies your exact tweet in their own Twitter stream, so that their followers see it as well. Using this shorthand will encourage people to click your link and retweet the petition so their followers see it.
With only 140 characters available, you can't afford to use 20-30 characters just on the link to the petition. Free link-shortener websites like bit.ly work great. (MoveOn's petition website automatically generates a shortened link when you post to Twitter from the "Share" button on your Dashboard.)
You can tweet the link several times a day with various hashtags to make sure people see your petition and are kept up to speed on your progress.