It's not possible to Bookmark songs or artists, but you can still have similar functionality by using Likes. You can Like songs, artists or albums.
Please note that you can like a track *without* giving it a thumbs up. To do so, click on the name of the track, then click on "Like this track."
Likes, along with your previously Bookmarked songs, will automatically show under My Profile on the left side.
You can still create new bookmarks on mobile and home-listening devices, which will also show up under My Profile when signed in to Pandora on your computer.