“Share” is a new item in the right click menu on 10.10. It contextually works for both files and document fragments (text, images, links, etc), and by default includes a few items like sending a via file Airdrop, adding to app libraries (mine has both Aperture and iPhoto listed), posting to Twitter, etc. I can see this being hugely useful if somebody whips up some simple Share items for uploading to a pastebin site, generating a Github gist, or so on.
“Actions” are a contextual hover menu added whenever an appropriate element is selected in a document. Each Action can open its own sub-window and perform actions on whatever the selected thing is, then send those changes back to the main document when the changes are done. For example, 10.10 is bundled with Markup, which allows basic editing of any image in a rich text field without needing to open a separate program. Some basic Actions that come to mind that could be really handy in this context would be code reformatters or converters.
Combined, these two UI elements are basically the successor of the Services menu. The biggest thing here is that it looks like both of them will be able to be installed and managed via Mac App Store downloads, rather than the manual wrangling of scripts or Automator files that’s kept the Services menu off the radar for most users.
(It would be really nice if the Services menu itself were also supported, but I suspect Services is likely to get deemphasized or gradually phased out once the new Extensions framework starts picking up speed.)