To the left of each book by default is an empty checkbox. When you click that, it by default marks it as “read”. It also pops up the book manage screen like below:

If all you want to do is mark a book read, you can close this popup. However this also gives you the option to:
– Change the status from “Read” to “Did Not Finish”, “Want To Read”, or “Not Interested”.
– Reset the status of the book to unread by clicking “None”.
– Make any private notes about the book that are only viewable to you.
– Change the date you read the book.
– Give a rating of the book.
– Write a review of the book, which will be shown on the books page.