Old Timer refers to someone who has been involved in a certain position for a long period of time who believes they are an "Elite" of sorts, thinking they have knowledge of everything, and that they are somehow superior than newer users.
For example, I was an Old Timer of various Garrys Mod servers, Forum Groups, and writing groups. The Writing Group is what made me an "old timer" class individual, constantly believing I was above newer members. The Forum Group only solidified this belief, making me have the mindset that I was a dictator of sorts. I was promoted to a Moderator of the group, which boosted my ego, making me feel even more like the Leader, and considering I was the last "Oldie" member left, I felt like I was the supreme master of the group.
The group died off shortly after. Everyone moved on to more productive things. I did too, and I really regret falling into that kind of mindset.
TL:DR, an Old Timer means someone who thinks just because they have stayed in a position longer than you means that they are always going to be superior to you in every single way.