Personally as an absolute basic laptop id say:
1.7 Ghz processor
Internet/wifi connection
dvd writer
40 gig harddrive
Memory for an XP computer - ideally 512 mb upwards
If Vista Basic, although min is 512mb, iid go for at least 1 gig, although ive heard it runs "just right" on 2 gig
You could probably get away with a much less powerful computer, but its easy to get a laptop/computer and then want to be able to do more stuff on it, so id at the minimum go for the stats above, and if your buying new, these will probaly be the absolute mininum anyway
Id highly recommend aginst going secondhand, can be too many problems, and something to consider is laptops have a "hinge life", basically they have a certain amount of times, before the hinges start going on them, which isnt good
AS a laptop itself consider
Battery Life
Weight
Hinges (play about with opening closing, do the hinges feel stiff?)
and durability - I.e fujitsu laptops are fantastic, but cosmetically they are the worse ive seen across most models of fujitsu and paticulary the hinges.
Personally id say, write down a list of what youw ant out of the laptop, things like, are you going to play the latest games? are you going to do this or that. Then go to pc workl with this list, get a couple of prices your happy with, go home compare to the top ten retailers, if cheaper online, print out the price, head to pc world and they WILL price match it, and give 10%
One more thing, if you feel pressured into buying the most fantasic laptop the guy has to offer....then thank him for his time and walk away, you do not need the best laptop ever, you need whatever suits your needs.