In order to answer this question you need to understand how Twitter works. The following article summarizes Twitter, how to set up an account and how it can be used to help bring attention and visitors to a website.
What is Twitter?

Twitter is a social media platform used to deliver short summarized statements (140 characters in length) of just about any type. Generally, Twitter is used to generate conversation on a particular, brand, person, or anything that can be expressed. Its marketing impact is significant in that it can be used to quickly generate interest in a topic reaching both regular followers and other interested parties in a very short period of time. Twitter has also added the ability to include a picture within the post.
How to set up a Twitter account - Instructions on setting up a Twitter account
Creating Tweets to bring traffic to your website
Twitter is short and to the point since you have only 140 characters to get your message to the reader. If your intention is to use Twitter to help drive traffic to your website, then your message should have that goal in mind. It should be a message that will annoy your viewers. It should be a message that will bring interest to your website. Information on your site, your cause, or related events should be highlighted. Here's an example:

Let's say that you have an astronomy enthusiast website named StarGazingFans.com and you want more traffic. Plan events and post articles that will drive people's interest to your website like:
Post tweets regarding current events or articles related to professional culture for your interest. It's also important to keep your twitter feed lively and interesting so that people will want to see what you have to say. You can use humorous posts or pictures of events to accomplish this goal.
How to be successful with your Twitter account
Here is a list of suggestions that will help you use your Twitter account effectively:
- Be consistent with your message
- Make sure your 140 character message is expressive
- Post daily or event appropriate tweets
- Learn how to use a hashtag to make it easier for your posts to be found
- Follow people and post daily to them or retweet from them
- Follow people who think you're interesting and have a lot of followers
- Use your tweets and interactions with other tweeters to show that you care
- Use other resources to find Tweeters who fit your targeted demographic. For example: peekanalytics.com
- Hire a Twitter expert or social media manager to improve the management of your tweets
- Develop a friendly voice and tone with your tweets - don't shut other people out where possible
- Build relationships with your followers and those who express interest in your posts
- Use Twitter to deliver coupons, sneak previews or links that will give your users insight to an occasion on your website
- Encourage people to retweet your information or messages
As you increase and refine your audience to match your website's demographics, you will find that you are strengthening the core group of people who will have interest in your website. The people will act as the foundation, providing a steadier rate of website traffic that you can build upon.