The record or Arm button is a function found on recorders and not necessarily mixing consoles. This button’s purpose is to determine if an individual is going to be recorded when the system is put into record. If a track is not record enabled and the system records, nothing will be recorded on that track.
This concept makes more sense if you think about a stand alone recorder like a 24 track tape machine or 8 track ADAT machine. If you want to record drums to channels 1 through 4 you would only arm those channels and then put the machine into record. Once those tracks are recorded you would turn off the record button (or disarm) on tracks 1-4 so that they play back what you have recorded. Now you record enable or arm track 5 to record bass while tracks 1-4 are not armed and playing back.
The record button seen here (and every other page of this website) is similar to those found in software DAWs. These programs have a software mixer that is a combination of an audio mixer and a recorder. In addition to mute, solo, and fader controls each channel represents a track of a digital recorder (built into the program)