The highest version of DirectX supported by the card.
Does It Matter?
Absolutely. This number gives you a very good big-picture view of a video cards capabilities. If you have any intention of playing games with your video card, you'll want to pay careful attention to this number when making a purchase.
Video card manufacturers typically release an entire range of cards supporting a given DirectX level. But there will be older cards on store shelves which won't support the latest and greatest. Be careful you don't get stuck with something missing the latest capabilities.
DirectX is an API created and maintained by Microsoft for the purpose of programming 3D applications. With different versions of DirectX new capabilities are added, and the minimum requirements for support go up. For example, version 9 of DirectX introduced pixel shader version 2.0 support, so in order for a card to be DirectX 9 compliant, it would have to support PS 2.0 (as well as other requirements).
Game developers target their games to these DirectX levels. Often they'll create multiple different modes to support different versions of DirectX and thus different generations of video cards. Older cards which don't support the latest DirectX version often have to use downgraded visuals in order to get a game to play.