Dragon Riders are humans or elves chosen by unhatched dragons as their Rider, forming an immensely powerful and unbreakable mental bond between the dragon and their human/elf. Often, they communicate telepathically between each other and as their bond grows stronger with time, their souls and minds become so synchronised they are almost a single being - thinking and acting in accordance to the other's thoughts.
This bond with their dragons resulted in Riders to gain the ability to use magic. This is done by uttering several words of an 'ancient language', and using one's energy to feed the magic's power.
In the book series, the main character (Eragon) and the main antagonist (King Galbatorix) are the only remaining Dragon Riders (to tell you anymore would mean spoilers).
Joseph: I know man, they are gonna try to mosh in the pit
Jared: its okay, they are gonna trip and fall on there trip pants.