Non-US people in discussions about TV series and reading TV history may be wondering: What is "syndication"?
"Syndication" is a term closely tied to US television and the mostly US-specific concept of "network television".
It's a large country that was early into television, and early into commercial television, and as a result they have a very complex and decentralized system of TV broadcasting.
People actually from the US feel free to correct me on anything I'm saying, I'm not in the business and I'm not even a consumer, I'm just piecing this together from cultural osmosis from across the seas and from wikipedia articles like en.wikipedia.org/wiki/Broadcas… .
In the US, television was originally run by many local TV stations run independently. Over time they started affiliating with each other into TV "networks". This mainly meant networks in an organizational sense, rather than in the sense of a physical radio network of relays all broadcasting the same signal. In fact, as the phenomenon of local broadcasters was considered a good thing and something important, regulation was put in place to make sure a limited number of hours were controlled by a centralized entity, the other hours were supposed to be managed locally.
Syndication is kind of a way around this. Instead of simply relaying programming from a centralized broadcaster, which is restricted to certain hours, local stations that are part of a network can buy recordings of TV series through their network. This does not have the same restriction, as the decision on what to show when is made locally, even if it's centrally-produced content.
When a big broadcaster produces a TV show of their own, they might first air it on their own national direct-relay hours, but after this exclusive airing they might sell it to their network of local stations or even to other TV networks, for local stations to buy and show on their own time.
This is syndication, and this is the main way reruns happen, which is what people are referring to when they are saying things like "after its original run Star Trek gained more and more of a following during its years of syndication". It means local TV stations were showing Star Trek episodes on their own time.
Again, any insight from US people welcome. The above is my impression of how this works, not the ironclad truth.