The DS Exit is a completely different game from the PSP and XBLA versions. It's more like Exit 1.5 because it incorporates some of the features from Exit 2 (which you can get on XBLA or if you're in Europe, on PSP) but the levels are completely new and original.
To answer the OP's question, Exit is a puzzle/platformer in which you have to manipulate your character, Mr. ESC, through various dangerous locales to rescue people in need. You then have to use the abilities of the rescued persons to get everybody safely to the EXIT door. For instance, Children can crawl through tight spaces and can run across crumbly floors without breaking them, but have a hard time climbing up or down gaps and they can't move heavy objects or swim in deep water. Fat people can move extremely large blocks around but need a lot of room to move and although they have an easy time dropping down large gaps, they need help to get back up. And so on.
Honestly, the Exit games are some of the best puzzle games I've played in a long, long time, and it's well worth picking them up. I paid full price for my copy of Exit DS and I haven't regretted it for a second. Just don't try to use the touch screen controls, they're awful.
I wrote a longer review over at Destructoid
if you're interested.