There's a few varieties - you can get an overview of them here - but I'm just going to cover the most rudimentary. They can give people a much richer multi-media experience, with added video and audio, product content and the like. They give the consumer a little something extra and link directly through to your content.
a surprising little something extra
I have this basic version (a Summary Card - which has description, title and a thumbnail) installed on this blog. If you do nothing else, do this. The code is pretty basic. I'm just using a simple site overview across every page. Just edit the bold bits accordingly and add it to your /head tag:
Once the tags are live, you can test the card with the official Twitter Card Validator. If you have a blogger blog (like this one) just go to Template, then go to Edit HTML, then cut and paste the above into the /head field. I hosted my image over on good old http://s809.photobucket.com.
Once it's all up and running and validated you can keep an eye on your analytics by being logged into your Twitter account and going to https://analytics.twitter.com, then clicking on the Twitter Cards tab at the top.
If you want to see it in action, share this post on Twitter then click it to expand it once it's posted. Go on, you know you want to ;)