In most
graphics software, "alpha" is the value of
opacity for an object or color. If something is 0% alpha, it is totally
transparent; if it is 100%, it is totally
opaque.
This is usually implemented by creating a special alpha channel-- in other words, thinking of Alpha as the fourth primary color. Most image storage mechanisms already work on the idea of three channels, for red, green, and blue. Each channel is essentially a separate black and white image, and so you have three black and white images, each one storing the amount in each pixel of red, green and blue respectively. An alpha channel is simply adding a fourth black and white picture that stores the opacity level of each individual pixel.. and so each pixel is thought of by the computer as a mixture of red, green, blue, and alpha.
PNG has an 8-bit (256 level) alpha channel, and so it can have parts of the image be partially transparent with different amounts of transparency than other parts of the image. GIF has transparency, but it's 2-bit and isn't really any real usage of the idea of alpha; GIF just selects one specific color to be the "transparent" color, and every pixel of that color is wholly transparent.