I listen to aircraft land and take off weekly. I noticed in taking off they usually only use about 3 to 5 different directions. I don't understand what they mean when for example they said to the pilot, "American 1450, ok for take off 27 left, head 265" The only part I am concerned with, the part I don't understand is the heading part. What does the 265 represent? The next plane was told to line up and wait. Shortly it was told that it was ok for take off, but to head 235. Please explain.
What you are hearing is a departure clearance. ATC is giving instructions to pilots on the ground before they take off on what runway to use and what to do next.
27 Left is the runway, which points close to 270 degrees. We shorten it to 27 because although it may not be exactly 270 degrees that is close enough. 27 Left means there are more than one runway in parallel, if there's a 27 left then there's a 27 right.
The second part of the clearance is the departure heading. 265 is very close to the runway heading, so ATC is telling the pilot to maintain runway heading. The second flight ATC is telling to make a left turn to 235 after departure, this is either to get the flight closer to its destination heading, or more likely for spacing to keep the flights separated. There was probably an altitude instruction in there somewhere, like 'climb to 2000ft, then turn left to 235.'
Depending on what direction (destination) the airplane will be going to, initial headings may be assigned to put the airplane in the appropriate traffic flow to best serve Air Traffic Control routing needs.
"Head 265" (probably actually phrased "fly heading 265") means that "after takeoff" Air Traffic Control is instructing the airplane to fly a heading of 265 degrees. Further routing instructions will be given to that airplane later.
It's a bit more complicated and involved than my basic explanation above but I assume you are quite unfamiliar with how Air Traffic Control routing works so I am trying to keep my answer very basic.
Hope this helps.