If you’re looking to create snowfall over a still photo or moving video in After Effects with no third party plugins, try CC Particle Systems II. The Faded Sphere particle type is ideal for simulating snowfall.
Simply set the birth rate, longevity, and emitter position and size for your particles. Then, set the particle birth/death size, variation, and coloration. In the Physics section of the effect, I recommend using either Twirly (for a windy day) or Direction animation setting (for more vertical snowfall when direction is set to 180). Play around with the velocity, gravity and extra option. Have fun!