Also known as Topics for Objects. 

Released in Spring '14, Record Topics are set to replace public tags - Hooray!

http://docs.releasenotes.salesforce.com/en-us/spring14/release-notes/rn_general_topics_for_objects.htm

See this good post showing an overview of the new feature Using Topics to Organize Records.

The Good

• Easy to use once set up. 
• Can use in list views, which is excellent.
• They work with groups. Add yourself as being a knowledgeable person about a topic
• They are an automatic "chatter group" showing records and discussions around that topic. 

The bad

• Is there really much benefit over the dreaded Multi-Select Picklists?
• No reporting! Come on! What is the point of views if you can't also use this functionality in Reports.
  • Well reporting was released in Winter '15 but it is next to useless! It only gives the prefix of the object that topics are assigned to - not a link or any details about the record it is assigned to. 
• No formulas! Stupid!!! 
• No API! So it looks like there is no way to bulk assign topics to records via the dataloader or another tool (see http://docs.releasenotes.salesforce.com/en-us/spring14/release-notes/rn_api_meta.htm - no mention of topics). 
• Not available to workflows or triggers.
• Adding a topic in a chatter post does not add it to the record - topics in posts and topics on records seem to be completely different.
• Note that setting this up will delete your public tags, if you have used them, though I don't know why you would have. 
  • See Migrating Public Tags to Chatter Topics on Records in Salesforce for a nice solution to this
• You can't seem to modify the fields in the list view on the Topics Record Tab (as shown here) and there does not seem to be an idea for it. 

The Annoying

• The UI on the page to set them up is terrible. I can bet you that you will miss the save button at least once. 
• You have to set this up for every object on every org (except ones created after Spring '14). 

You can't use them for

• You can't use them to auto add records to campaigns because there is no workflows or triggers on them. 
  • You can do a view and manually add that view to the campaign. 
• You can't use them to auto populate Mailchimp or Campaign Monitor lists because they are not a field that you can filter on. 

Verdict

Like Tags, they look good, they seem good, but they are totally useless - at the moment. I hold out hope that they will be much better in the future. 

Articles