Anti-aliasing is the process of 'smoothing out' the colours of an image - in its simplest form, taking two pixels and moving the RGB values of each closer to the other.
The 'Blur' tool of many more comprehensive art programs performs this function, although it is a very blunt instrument.
A more refined approach is required with pixel art, where the aim is to reduce the 'jaggedness' of the lines where two shades meet.
NOTE - Anti-aliasing is not a holy grail of pixel art. You may not even need it. There are many situations however, where smoothing out jagged edges with a pixel gradient improves the appearance of a sprite.
This method first lightens the colour of the line, then shade away from each 'step' in shades, getting gradually closer to the background colour.
This method is most useful for shading towards the outlines of sprites, though it need not use as many shades as this example.
NOTE - Do not anti-alias outside the outline of a sprite (ie - in the transparent bit) unless you anticipate placing it against an unvaried background.
This is an alternate, slightly different method. The RGB value of the line remains unchanged, but smoothing of tone is focused more at the 'joints' or 'steps' of the edge.
I prefer this for drawing lines on a surface - tattoos or paint lines for example.