Not all stalls start from the trailing edge and progresses forward. In general, three types of stall are described:
Trailing edge stall- This is the preferred stall characteristic; The turbulent separation point moves forward from the trailing edge with increasing angle of attack. usually found in airfoil sections with thickness ratios 0.15 and greater.
Leading edge stall- The flow separates abruptly from the leading edge without subsequent reattachment; usually found in airfoils with thickness ratios between 0.09 and 0.15.
Thin airfoil stall- Found in airfoils with small thickness (< 0.09), the separation occurs at the leading edge and the flow reattaches at a point which then moves progressively rearward with increase in angle of attack.
Buffet is a kind of vibration caused by aerodynamic excitation, usually associated with separated (or turbulent) airflow. As the aircraft approaches stall, the airflow over the wing becomes turbulent and if it flows across the horizontal stabilizer, buffeting may occur.
Whether low speed (stall) buffet happens or not depends on the aircraft characteristics; for example, the stall starts from the root in case of straight wings and the stabilizer is affected by the turbulent airflow before the outer wing stalls- this can be used by the pilot to take corrective action (as @Ron Beyer points out in his comments). In case of swept wing aircraft, the stall progresses the other way around- from tips to rot and it will be difficult for the pilot to use buffet in any real sense.
This buffeting can act as a warning for the pilot that the aircraft is approaching stall and he/she has to take corrective action. In aircraft that do not show this behavior, some warning cues, like stick shaker are sometimes incorporated in order to warn the pilot.
As density altitude increases, the angle of attack required to produce turbulence at the top of the wing (stall angle) decreases till a point is reached that the high speed buffet (Mach buffet, due to supersonic airflow) and the stall buffet (as already explained) converge. This point is called the coffin corner.
"CoffinCornerU2" by Department of Defense - AF (C)-1-1, Flight Manual, Models U-2C and U-2F aircraft. Page 6-11. Licensed under Public Domain via Wikipedia.
From FAA Handbook FAA-H-8083-3A:
An airplane’s indicated airspeed decreases in relation to true airspeed as altitude increases. As the indicated airspeed decreases with altitude, it progressively merges with the low speed buffet boundary where prestall buffet occurs for the airplane at a load factor of 1.0 G. The point where the high speed Mach indicated airspeed and low speed buffet boundary indicated airspeed merge is the airplane’s absolute or aerodynamic ceiling. ... the airplane can neither be made to go faster without activating the design stick puller at Mach limit nor can it be made to go slower without activating the stick shaker or stick pusher. This critical area of the airplane’s flight envelope is known as “coffin corner.