There are two ways to vote. When you visit the home page (or click the Litmocracy icon in the upper left corner of this page), you’ll see the heading “Some pieces you haven’t voted on yet.”
There are boxes with the beginnings of the pieces presented, and in the bottom of each box there’s a “read more…” (or maybe “read and vote…” - I can’t remember) link. If you click this, then you’ll get a page that has that whole piece on it, along with another one below it from your list (if you’ve voted at all). In between the two are some buttons you can use to indicate which one you like better. Clicking the buttons brings another piece from your list to that lower section. The site keeps presenting pieces from your list until it has pinpointed where in your list the new piece (always displayed above) should go, and then it puts it there.
On the home page, between the boxes with pieces you haven’t voted on, there is a picture of a ballot box. You can click on this picture to go directly to your list for the category that the piece above it is in. This allows you to read the full piece and then use QuickVote to indicate where in your list it should go.
I like to use the step by step method (“Read More…”) to re-familiarize myself with the other pieces on my list, but if I just did it recently, then I’ll visit my ticket (by clicking the ballot box) and add pieces that way as I read them.