Ok, so I have an explosion scripted with smoke effects, however the smoke (an image on a white background) is produced as a square with the white background. How do you render the particle without the background?
That’s because your texture doesn’t have a transparent background. There are a few ways to get that:
- Have an transparent background / alpha channel in your texture. That way Unity will make it transparent.
- Use the image’s grayscale as alpha (there’s a checkbox for that in the texture import settings)
The best way for you would be to get a transparent texture 
Like this one: http://read.pudn.com/downloads65/sourcecode/game/232453/smart_version/media/materials/textures/smoke.png
Also note that you must use a shader that supports transparent textures.
As you’re doing smoke, I’d suggest the bundled one: Particles ->Alpha Blended or Particles → Multiply.