Ancient is a casual constructed Magic: The Gathering format that only allows cards from the original border sets (Alpha through Onslaught).  In contrast to QL Magic, Ancient uses modern rules.
While the card pool is limited, it also allows players to use cards from Magic's past that might be deemed too weak by today's standards - creatures in particular. There are SOME exceptions where cards from a cycle that were printed in a 'Modern' set are allowed, but only when these cards were printed to complete a cycle from an earlier set (most notably, the Zendikar fetchlands that completed the cycle from Onslaught).
Reprints from the Alpha-Onslaught sets with new borders are allowed. 
Side note: a group of players in the Netherlands also play a format they call Ancient, which is completely different.