Provided that the image (usually a “clip art”) has a color that can be easily assumed to be the background:
convert input.png -transparent white -fuzz 90% output.png
Replace “white” with the color you want to replace, or “rgba(0,0,0,0)” to set it to a custom value.
“fuzz” is the tolerance, that can be tricky to set with images with a drop shadow. For the average antialiased PNG image can be around 90%.
Being ImageMagick, the input can be any image format, but lossy compressed JPEG images tend to produce bad results.