Celerons are ok for computers that are used in offices.
a 2 Ghz Celeron is roughly equal to a 1.4 Ghz Pentium 4.
So basically Celeron processors are for price conscious people who don;t use their computer for anything other than word processing and other tasks that don't really need a lot of brute processor power.
For graphic intensive applications, Celerons = Crap.
As for AMD Barton processors, a 2500+ barton (running at 2 Ghz) is roughly equal to a 2.4 Ghz pentium 4 in terms of performance.
Bear in mind I'm pretty clueless about hardware on the whole. But I guess Pentium is just superior technology, more performance with less power lost to heat. So it's probably better for mobile devices where you work on a battery and and less noisy assuming you have a cpu fan for cooling.
Athlon is similar, it's powerful enough to compete with Pentium but it's less efficient and overheats more easily. At least this was true a couple of years ago, now both manufacturers have evolved with new mobile technology etc, so it may have changed.
Obviously, you don't get hyper threading with a Celeron (or an Athlon for that matter).
No, I mean heat issues. AMD processors have always outclassed the Pentium at a given frequency but they do produce more heat and are more vulnerable to overheating. Do you remember that experiment when a guy ran a P4 and an Athlon both at 100% load then removed the fan? The Pentium was damaged but the Athlon actually melted..
But again, it may have changed the last year or two for all I know.