Klout is a service that uses social networks to measure your influence over others. Scores are tabulated for each user on a scale of 1-100. The higher your score, the more influence you have, according to Klout.
Twitter.com doesn't display Klout scores, but if you use Firefox or the Chrome browser, it's easy to remedy with add-ons. If you use Firefox, install the Firefox Klout add-on. If you use Chrome, install the Chrome Klout extension.
The next time you visit Twitter.com, you'll find Klout scores to the left of the Twitter handles.
Clicking on the scores will open a new tab with more Klout details about the user.
That's it. Now you can view the Klout scores of everyone on Twitter without having to leave Twitter.