From my (Catholic) understanding, Apostolic Tradition, in a nutshell, has to do with the Apostles "handing down" their learnings from Jesus to others.
As an example, at the Last Supper, Jesus broke the bread (I'm paraphrasing), gave the blessing and said, "...do this in memory of me." In Acts, the Apostles came together several times to "break bread" while rejoicing in the Resurrection. Some Christan sects do this once in a while. Others do it daily.
Another Apostolic Tradition might be, that the Apostles went from place to place, orally teaching the "Good News." People have continued to do that for the past 2000+ years.
All in all, it pretty much has to do with mimicking what the Apostles did.