DS = a portable games machine for playing portable games.
PSP = a handheld thing to play games that are sort of like PS2 games.
The DS has loads of great games designed for it, designed to be played anywhere, and designed for a quick go, whereas the PSP games seem more like console games stuck onto a portable machine, they haven't really thought about how people use portables, or done anything innovative with the interface.
Also, the newest PSP has made the stupidest ever design decision of only allowing games to be bought as downloads – meaning no second hand games, whereas DS games you can almost always get for a tenner 2nd hand.
For children it's a no brainer – almost all the kid targeted games are DS only nowadays (like the dog / cat / sims style ones they spend hours on). To be honest, if you're a busyish adult, it's a no brainer too – full sized console games are just too big of a time sink compared to a quick hit of DS.
I was in Smiths last year just after Christmas – saw people returning PSPs that they'd bought for their children who wanted a DS, they'd thought it was more expensive and made by Sony, so it must be better.
Joe