Most of the users do it for fun and to meet & chat with other people, some users organize full marketing campaigns on the social networks and use this one for audience and clients gathering. In any case, interacting with other Twitter uses is almost mandatory in order to meet the full social and full features of this website.
The most important thing you must remember when you are using Twitter is authenticity. In 2 words: be yourself. Twitter is a microblogging website designed for people interacting with other people, the result is a community of men and women talking about whaterver they like or love. Repeating all the time what others say, without adding nothing personal won't help in any way getting followers. Everybody has something to say, there are listeners for everyone. So start saying something... or, in this case, write tweets. Don't be very formal. Experiment and be casual.
Avoid mundane tweets about what you ate at lunch or dinner. Or about the weather in the winter mornings. Nobody might care... write instead about your passions and share your activities in a way will make people listen. They will definetly react to that. If you represent or are a part of an organization you can always write about stories (more or less funny) with people from the team.
Have an opinion and write it down. In the worst case scenario, when you are wrong, you'll find people that can and probably will correct you - you may learn something new from that.
If you are sharing information try to write the tweet as a headline: short, to the point, in the most interesting way you can think about. When reading a newspapers people are looking at the headline first, often your tweets are the headlines of the message you are sending.
Use hashtags to share, spread and organize information. But don't use them without reason and without knowing exactly what you want or they will lose their meaning. Use them strategically and appropriately.
Bring (push forward) people behind the scenes. Let's say that everybody knows or they will find out about the event or the information you are sharing using Twitter, but (almost) anybody knows about the people that organized that event or who produced the information. Bring them into others' attention. Even more, ask your follower about their oppinion. Twitter is a great method to get feedback from others, and you always need it.
Respond to other people's tweets. One of the greatest things about Twitter is that you can easily start a dialog with other users about (almost) everything.
Retweet others' tweets if you consider them interesting. Help them spreading information. They usually consider this action
The image was found here.