To help illustrate what an alpha is let's imagine it as a separate image from the main image. The alpha channel is an image that describes the transparency of the main image. The white parts of the alpha image describe the most opaque parts of the color, where as the black parts describe transparent areas of the color.
Grey areas are semi transparent.
Photoshop allows for multiple alpha channels in a single image. The alpha channels can be used to save and reload selections. They can be manipulated to get different effects. When an image is saved with an alpha channel, it is usually referred to as a 32 bit image.