It's called alpha blending.
In psuedocode, assuming the background color (blend) always has 255 alpha. Also assumes alpha is 0-255.
alpha=argb.alpha()
r = (alpha/255)*argb.r() + (1 - alpha/255)*blend.r()
g = (alpha/255)*argb.g() + (1 - alpha/255)*blend.g()
b = (alpha/255)*argb.b() + (1 - alpha/255)*blend.b()
note: you probably need to be a bit (more) careful about floating-point/int math and rounding issues, depending on language. Cast intermediates accordingly
Edited to add:
If you don't have a background color with an alpha of 255, the algebra gets alot more complicated. I've done it before and it's a fun exercise left to the reader (if you really need to know, ask another question :).
In other words, what color C blends into some background the same as blending A, then blending B. This is sort of like calculating A+B (which isn't the same as B+A).