In normal human words, an API is a thing that automatically returns info or changes info in a database when you ask it. They can be public or private.
Take the list of threads on the newschoolers form as an example. You've got a database with what is essentially a spreadsheet (or table) of thread subjects, times they were created and number of responses and views.
If the API is public, you can ask it "give me a list of the subjects with the number of views". Or "give me the 3 most recent subjects" etc.
Once you have the raw info, you can display it however you want (in blue or red or whatever).
The API might have the ability to return the information is a lot of different ways or it might only let you ask a single way. Kinda depends. You can also tell the API to change things in the database (table/spreadsheet). That is what happens when we post. It increments the reply, adds the info you typed, etc.