What are the differences in door types in a commercial airliner?
The only formal classification of door types that I could find is in 14 CFR 25.807, which specifies the emergency exit requirements for transport category aircraft. It defines Type I-IV, Type A-C, ventral and tailcone exits. The doors you usually enter and leave the aircraft through are probably type I or A:

Type I. This type is a floor-level exit with a rectangular opening of
not less than 24 inches wide by 48 inches high, with corner radii not
greater than eight inches.
Type A. This type is a floor-level exit with a rectangular opening of
not less than 42 inches wide by 72 inches high, with corner radii not
greater than seven inches.

These definitions are quite generic, but Transport Canada has a handy list of Aircraft Exit Profiles that gives specific details (often including the type) for many common aircraft. It says this for the 767 main doors:

B-767-200/300ER (Doors 1L/R FWD, 2L/R AFT)
Note: Some models equipped with 2 additional type 1 exits L/R just forward of wing.

H - 1.88 m (6 ft 2 in) W - 1.07 m (3 ft 6 in); Type A
Equivalent force required to move door control handle from closed/locked position to open position (Normal and Emergency): 12 kg
(26 lbs).
Approximate force required to open a door (lift upward) from unlocked to open: 18 kg (40lbs).
&quot;Counterbalance&quot; opening design.
Inward/upward opening doors.
Manually operated except some models equipped with electrical (normal operation) opening/closing option (FWD Entry Door).
Unique arming/disarming mechanism (lever, indicators).

Release button must be depressed and held to move arming lever to &quot;Slide Armed&quot; mode.
Yellow plastic &quot;Emergency Use Only&quot; (bendable) flag moves upward in front of Door Control Handle when slide is in armed mode.
Double channel slides.


Upward rotating Door Control Handle.
&quot;Uplatch&quot; mechanism holds door in open position. Button must be depressed while door is lowered initial few inches, to release.


You can see from that list that the details can vary a lot, even if the basic designs are very similar.
Coming up with pros and cons for every variant would be tough (especially for me, since I know nothing about it!), but obviously some aircraft are larger than others and that will limit how much space is available for the door mechanism. There are also operational considerations: aircraft that frequently operate out of smaller airports without jet bridges may need to have airstairs, which are usually built into the doors.