For your computer to have DirectX 11 support, you need two have components: the software, and compatible hardware.
The DirectX 11 API (the software component) comes by default with Windows 7 and 8, or if needed, you can find information about installing it manually at this page on Microsoft's support site. You can check the version of the DirectX API installed on your computer by looking at your dxdiag. It is near the bottom of the System tab, listed as DirectX Version.
The hardware component refers to your graphics card. Cards older than Nvidia's 400 GT series or AMD's HD 5000 series will come with an older version of DirectX and will not be compatible with DirectX 11, even if you have it installed. If you're not sure which version of DirectX your graphics card supports, you can, as with the API version, check on your dxdiag. It will be listed as DDI Version on the Display tab. If your graphics card does not support DirectX 11, you may want to consider upgrading to a newer card.
If your graphics card supports DirectX 10, but you have the DirectX 11 API installed, you still ultimately only have DirectX 10 support. Likewise if your graphics card supports DirectX11, but you only have the DirectX 10 API installed. You will always only have support for the older version between the two.
But I've played DirectX 11 games with a card that only supports DirectX 9 or 10 before!
Some games have built-in fallback support for versions of DirectX older than their recommended version. Essentially, this means that when the API detects that you have a graphics card that only supports an older version of DirectX (for example, the game recommends DirectX 11 and your card only supports DirectX 10), it will fall back to something that the older version of DirectX can interpret. This support for older versions of DirectX is something that developers must add manually, and will, therefore, not be present in all games. This is a possible explanation as to why you are able to play some, but not all DirectX 11 games.