Simply put, no - you can't play 3DS games on your DS system. In fact, even though 3DS games are the same shape and size as DS cartridges, they have a little tab on the right hand side to prevent them from being inserted into a DS. Even if you were to file this tab off (as some people on the internet have done), your DS is just incapable of reading the game, and thinks there's no game inserted. It is possible to play DS games on your 3DS though - so you don't need to worry about not being able to play your old favourites, should you choose to upgrade.
The 3DS is basically the DS 2, with a 3D screen, and is a completely new - and more powerful - system, and comes with a whole load of built in features before you even get to the games. As it's the successor to the DS, as time goes on more and more games are going to skip the DS and go straight to the 3DS - so you may find yourself needing to upgrade your system to play the latest Layton or Brain Training game.