I have hit a door with the inside of my left hand. I have a dark bruise on the inside and swelling on the outside and the side of my hand and is very tender. I can move my fingers though. What have I injured?

I'm sorry to hear about the incident however the rule of thumb (no pun intended) in these kind of cases is:

If you think you've broken a bone in your body, you haven't. If it was broken, you would know as the pain would not be bearable and your hand would be twice its normal size.

Nonetheless the only way to know if everything is 100% normal with your hand is to get an X-ray but if you've written this question yourself, chances are your hand is not fractured, just sprained, hence the bruising.

Here's some stuff you can do to ease the pain and get your hand back on track:

-Rubbing castor oil where it hurts is rumoured to help (some say the only thing about this that helps is the friction itself and not the oil).

-Squeezing a ball gently but firmly several times per day can dramatically help your hand to go back to its normal state.

-If you see the pain is not going anywhere than you could ask for a cortisone shot at your GP.
This is known to help tremendously.

You can also check this video out where a certified doctor will clearly explain the difference between broken, sprained and strained:

I hope this helped!