I occasionally toss around terms like alternate and economy picking without thinking, forgetting that not everyone is a mildly obsessive guitarist. So, an explanation then.
Both relate to how you use a plectrum to pick a guitar. Normally, if playing on one string, you would play with alternating downstrokes and upstrokes of the pick. This makes sense from an ease of movement point of view. If you tried to play all downstrokes instead, for example, you would have to move the pick up again to be in the position for the next movement after every downstroke. Using both up and down strokes simply makes use of this movement. Because you alternate between the two, this is a very basic form of alternate picking.
The difference between what's called 'strict' alternate picking and economy picking comes when moving between strings. The alternate picker continues to move up and down alternately, regardless of their movement between the strings. So if they had just played a note on the D string (the 4th string of the guitar) with a downstroke, and were going to play one on the G string next (nothing to do with underwear, I'm afraid, but simply the next highest string in pitch, and positioned immediately below the D string physically) they would play it with an upstroke. What this means, you might have noticed, is that they would go down past the string to come back up again.
Economy picking says that this is a waste of effort, and so would play the same combination of notes with another downstroke (well, technically as part of the follow through from the original downstroke, but you get the idea.) This is because the basis of economy picking is to always move directly to the next string when changing strings, regardless of what you were doing before.
So, to play a string positioned below the one they're on, an economy picker would always use a downstroke, while to play one where they're moving upwards to get to it, they would play the first note with an upstroke. An alternate picker, by contrast, would keep going down-up-down-up regardless.
The reasons people (and by people, I of course mean myself) can't make up their minds between the two are that they, and slight variations on them, offer different advantages.
Economy picking has the advantage of economy of effort. You never have to go past a string to come back and play it, as you do with alternate picking. You can arrange things so that they barely take any pick strokes at all to play. Arpeggios, in particular, are easy with this approach (which tends to get called sweep picking once you're dealing with arpeggios) allowing you to do your best Yngwie Malmsteen impersonation. And yes, I have an Yngwie Malmsteen impersonation. I never claimed to have any musical taste. There's also an argument for saying that it produces a smoother sound than alternate picking.
Alternate picking has a few main advantages. Firstly, it's quite easy to keep track of. You don't have to work out what your next few pick strokes are, because you know it's going to be down-up-down-up... Secondly, there is an occasional tendency for economy picking to descend into mush, where you can't tell one note from the next. Alternate picking tends to give better definition to the notes, even if it is harder to do.
Currently, I'm on mostly alternate picking with the occasional sweep for arpeggios (and also some hybrid picking, which a fancy name for using pick and fingers together), but presumably you'll want examples you might have heard. So, some reasonably well known guitarists who...
...Alternate pick a lot: Steve Morse (Deep Purple), John Pettrucci (Dream Theatre), Al di Meola, Guthrie Govan.
...Economy pick a lot: Frank Gambale, Steve Vai, Yngwie Malmsteen