Real and complex fractals are those fractals that use deterministic algorithms based on real or complex arithmetic. The terms real and complex refer to types of numbers. Real numbers are numbers that can be distances or the negative of a distance. Complex numbers the numbers you get if you add the square root of negative one (written i) to the real numbers and then close the set under the normal arithmetic operations +,-,*,/. This class of fractals includes Mandelbrot fractals, Julia fractals, Newton's method fractals, Glynn fractals, and orbit-capture fractals.
Randomized fractals are fractals that use random (or pseudo-random) numbers in their algorithms. It is possible to randomize almost any fractal algorithm but this category includes only includes algorithms that use random numbers in their most basic form. This class of fractals includes iterated function systems such as the Siepinski triangle, random midpoint fractals like the plasma fractal, and iterated object placement fractals that make nice textures.
Discrete fractals are fractals that use deterministic algorithms based on geometry or finite structures like the integers (mod n). These include initiator-generator fractals like the Koch snowflake, cellular automata like Conway's game of life, and the sandpile automata, and L-systems which are models of plants.