Write catchy lead lines and link the targeted site. Please know that if you write catchy leads the article witch the link is leading to have to be also great and unique. If you make like: "The best offer of the year for free" lines and you link to a site where visitors only may buy "blue medical pills with a letter V" - the next time they will not click on your links. The hardest thing is to come up with unique ides and writing them down. If you made a great content (like hoe to set up a server) users will love it and they will link it. Clear? Cheers,
Get more followers and put your URL in tweets, and see what happen in the traffic stats. But yeah! as ARTidas has already said... use catchy lines
There's a big question to answer before really answering that question: what kind of quality do you want? If you're not concerned by quality, I find that being vague (like just posting a link with no text) works, but people tend to bounce. If you're doing something CPC behind it, it may not matter. If you're looking for quality, I find that using frequently retweeted hashtags that are relevant works.
If you have a large network like a big friends circle then they can follow you in twitter and this way you can have better traffic.
You will need two things to drive traffic from Twitter. Good content and a large base of followers. You typically can only rely on about a 3-6% (usually 2-4%) click rate on any tweeted URLs. So, if your follower base is 1,000 followers, you will be LUCKY to get 35 clicks on your links, and can expect even less. So, first try to build a large network of followers. With that in place, content will become king.
I agree with the other posts - the key with twitter is getting more followers, but you need to be sure that the followers are the of the type that you want. If your twitter followers are friends then they may not convert into sales...
Build a follower community that is real. The key is making sure your community is real and they follow you because they want to hear from you... There are many out there with big numbers but the turn out (clicks are very low)...
First of all, you have to have a huge list of followers that are real people and are interested in your niche. When you already built a decent number of followers and already build a decent reputation that's when you do all of the advertisement basics: Catchy phrase then URL, make sure the page it's linking to have really good content.