Anachronisms
Episodes from Season 1 to Season 3 feature rotary phones with clear plastic finger wheels. These episodes take place before 1964, when the plastic wheel was introduced. Before that, the finger wheels were black and metal.
IBM Selectric IIs are shown used in the offices. They weren't available until the 1970s. Mad Men takes place in the 60s.
Mad Men is set in the early 1960s; however, in several episodes, telephones are shown with RJ-type modular connectors, which were not introduced until the mid-1970s.
There is a scene where Don and Roger are using urinals in the Men's Room, and the urinals have a privacy divider between them. These did not appear until the 1990s or 1980s at the very earliest.
Continuity
The picture of Pete's wife is different in later episodes than from the pilot. The picture in the pilot is obviously not the actress who plays Pete's wife (Alison Brie) most likely because she was not yet cast.
Office doors and office locations are different in the pilot than in later episodes. Most of the offices in the pilot do not even have names on the outside yet.
Errors in geography
In many outdoor scenes, the tops of palm trees can be seen - Particularly in the horse-riding/stable episodes in the first few seasons.