Particle effects
In Team Fortress 2 particle effects serve as decoration or visual cues for certain in-game events. A detailed explanation of particle effects in general can be found on Template:W. Documentation for the Source engine's implementation can be found here.
Unusual effects
Unusual particle effects are particles attached to Unusual quality items.
There are 1008 possible combinations between hats and Unusual item particle effects (14 effects × 72 [current] possible unusual hats). The different particle effects are viewable in this animation, and here as they appear in-game; as well as in this video.
Currently, 22 out of 94 different hats cannot possess any of the 14 possible effects. These hats include: