Twitter, FaceBook, and many other sites are using OAuth or some variation for user authorization. With OAuth, rather than storing a username or password, you request a token from the website, which you shore and use to access the site in the future. If you’re not already logged in to that site, you’ll see a login page. If you’re already logged in, you’ll get a prompt asking whether you want to authorize that application.
For most sites, the authorization prompt has clearly marked buttons telling you to proceed if you specifically asked that application for authorization or cancel if you got there some other way. With Twitter, for example, the cancel & proceed buttons are red & green.
Flickr (which actually uses its own variation), on the other hand, gives you two paragraphs of text, which you have to read very carefully, and two identical next buttons. Not a good design.