How many difficulty modes does Dark Souls 1 have?

The "++" you are seeing are known as New Game Plus and the difficulty increase with each run through up to 6 times

After the ending of the game a new game will start. This is called New Game Plus (NG+). New Game Plus begins after you have defeated the final boss, Gwyn, Lord Of Cinder. The game begins again, allowing you to keep most of your gear and items, but with increased difficulty. After NG+ is completed, the game will continue onto NG+2 with slightly increased difficulty again. This continues all the way to "New Game Plus 6" (NG++++++)

Aside from enemies being stronger in general, the following are the changes in enemy HP and dropped souls between each new NG+

1st NG       Regular NG values
2nd NG+      Enemy HP and Souls increase by inconsistent amounts.
3rd NG+2     Enemy HP and Souls are 7% greater than NG+ values.
4th NG+3     Enemy HP and Souls are 10% greater than NG+ values.
5th NG+4     Enemy HP and Souls are 14% greater than NG+ values.
6th NG+5     Enemy HP and Souls are 19% greater than NG+ values.
7th NG+6     Enemy HP and Souls are 25% greater than NG+ values.

You can go up to NG+6 with a linear increase in difficulty after Ng+2, but otherwise there's no difficulty modes.

More details in the wiki.