The DS Lite does come with a power cord to charge the lithium ion battery. Out of the box, the DS cannot surf the web. It does have a chat program called Picto Chat that allows you to chat with other DS users in the area or over the web. The DS Lite can play DS games and Gameboy Advance games, but not Gameboy or Gameboy color games.
There is a cartridge you can buy that will allow the DS Lite to surf the web using the Opera browser which is endorsed by Nintendo.
There are also flash cartridges, R4 (www.r4ds.com) and Supercard (www.supercardnds.com) are a couple names, that will allow you to play music, videos, and homebrew games on your DS. These cards are not endorsed by Nintendo.
I'm assuming you want to do the pros and cons of a DS Lite to the PSP so here you go:
Pros
-Much longer battery life, up to 20 hours on a charge.
-2 screens that can view information independent of one another.
-Touch screen that really innovates game play
-Larger library of games
-Flash based cards have almost no load times
-Clamshell design is more rugged than PSP and resists scratches on the screen.
-Less expensive of the 2 systems ($129.99 to $169.99)
-Games are geneally less expensive as well.
Cons
-Graphics aren't as good as the PSP
-Does not have web or music functionality out of the box
-The DS web functionality only supports WEP encryption (not WPA or higher)
-Large volume of childrens games (could be a pro if you're buying for a child).
-Flash cards only hold about 1/14th of the space of a PSP UMD.
-PSP also has UMD movies
That isn't all of them, but it should get you started. I'd recommend reading the Wikipedia article on the DS Lite, it has a lot of good information.