Particles are just points in space, by default have no geometry or anything for light to interact with so they do not render. There are several ways to render particles, but for your case I would use Object particles.
First create the object you want to use for the snowflakes. The level of detail depends on how close and in focus the snowflakes will be, but for most applications a simple circle with 8 vertices should work fine.
Next go to the Render rollout of the particle system and select Object and select the snowflake object to use for the particles.