Merriam-Webster defines propagation as:
The act or action of propagating : as a : increase (as of a kind of organism) in numbers b : the spreading of something (as a belief) abroad or into new regions c : enlargement or extension (as of a crack) in a solid body
For our purposes, definition B is most appropriate. Quite literally, propagation is the time during which your DNS Zone information is spread abroad to servers that didn't know about it before.
Why does it take so long? There are various reasons, but some studies have suggested that there are literally millions of DNS Servers that need to be notified of the new information!
Propagation takes two forms, changes to your DNS Zone and changes to your WHOIS information. The WHOIS is the master record that tells every DNS Server in the world which is the authoritative server for your domain. A change to your WHOIS information (done my your domain registrar) can take up to 72 hours to propagate fully. A change to your DNS Zone information typically affects only a handful of servers, and so is done much more quickly.