Alternate Picking is a picking motion where the pick alternates between up and down and/or down and up. Let's start out with an example...
Now each note is to be picked the other direction:
So lets start with a down motion. You play the open E with a downstroke (picking the string with a down motion of the pick) and then on the first fret you pick upward. Then we would continue on the 2nd fret with a down motion, then on 3rd fret we have an up, 4th fret is down, and so on. So, you would then start on the 5th string with an up motion, keeping the chain of down, up, down, up, etc.
Let's take a look at this one, it is in triplets starting on a down motion alternating.