What are the differences between types I, II, and III emergency doors?
The major difference is size.
FAR part 25.801 defines them as:

(1) 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.
(2) Type II. This type is a rectangular
opening of not less than 20 inches wide by 44 inches high, with corner
radii not greater than seven inches. Type II exits must be floor-level
exits unless located over the wing, in which case they must not have a
step-up inside the airplane of more than 10 inches nor a step-down
outside the airplane of more than 17 inches.
(3) Type III. This type
is a rectangular opening of not less than 20 inches wide by 36 inches
high with corner radii not greater than seven inches, and with a
step-up inside the airplane of not more than 20 inches. If the exit is
located over the wing, the step-down outside the airplane may not
exceed 27 inches.

There is also a type IV:

(4) Type IV. This type is a rectangular opening of not less than 19
inches wide by 26 inches high, with corner radii not greater than 6.3
inches, located over the wing, with a step-up inside the airplane of
not more than 29 inches and a step-down outside the airplane of not
more than 36 inches.

If you read the full text, you will also find definitions for ventral and tailcone exits. For pictures I refer you to this answer.