Twitter can be great for promoting a website and getting a good amount of traffic but you have to do it properly. First, you don't want to spam or post a huge amount or Tweets, especially useless or junk Tweets. If you post thousands and thousands then people will just stop reading them because they know that you are just Tweet happy.
One thing that you want to do is Tweets some headlines and keyword for any articles / posts on your website with a link to the article. Also, include an image with the Tweets as that will get a lot more people checking your Tweets and possibly going to your site.
You also want to Tweet some stuff that is not something of yours so it doesn't look like you are just posting your own stuff and that's all you care about. Find some big websites, preferably that have a Twitter account with a lot of followers and send out a Tweet of something that isn't your about every 2-3 Tweets. You want to post things that people would be interested in.
When you do send a Tweet about someone else's content or article make sure to tag them properly so that you are giving them credit for what the Tweet is about and also to hopefully get some of their followers to follow you and/or read your Tweets. What you do is put a period right befor you put their Twitter username in the Tweet. Something like this ".@TWITTERUSER". That will make sure that the Tweet goes out to all of their followers also.
There are many ways that you can get traffic from Twitter but this is something that has worked real well for me and many others.
Cheers, Dean.
