TL;DR: The One Ring contains not only a portion of Sauron's life-force & power, but also the ability to control the other rings of magic given to each race.
There are already a lot of good answers here, but they all refer to the books & expanded lore. Since the original question was about the movie version, I thought I'd write up an answer using only what was there. In the first movie, there are two main scenes that explain the power of the ring.
The first is during Galadriel's voice-over at the beginning. After describing the creation of the rings and who received them, she states:
GALADRIEL (V.O.) : For within these rings was bound the strength and will to govern each race.
[FADE TO BLACK]
GALADRIEL (V.O.) : But they were all of them deceived.
[FADE UP: An ancient PARCHMENT MAP of MIDDLE EARTH...moving slowly across the MAP as if drawn by an unseen force the CAMERA closes in on a PLACE NAME...MORDOR.]
GALADRIEL (V.O.) : ...for another ring was made.
[TEASING SHOTS: SAURON forging the ONE RING in the CHAMBERS of SAMMATH NAUR.]
GALADRIEL (V.O.) : In the land of Mordor, in the fires of Mount Doom, the Dark Lord Sauron forged in secret a Master Ring to control all others.
[IMAGE: The ONE RING reflecting FIERY LAVA. FIERY WRITING emerges on the plain BAND OF GOLD.]
GALADRIEL (V.O.) : ...and into this Ring he poured his cruelty, his malice and his will to dominate all life.
[IMAGE: THE ONE RING falls through SPACE and into flames...]
GALADRIEL (V.O.) : One Ring to rule them all...
This tells us that even the basic rings of power have within them the "strength and will to govern each race." That is apparently an inherent quality in ALL of the rings, not just Sauron's. The difference is that Sauron's controls the other rings, not the races. The One Ring itself is a conduit through which Sauron can tap into the other rings.
Next, we have the scene where Gandalf informs Frodo that his ring is the One Ring:
FRODO : But he was destroyed...Sauron was destroyed.
[ANGLE ON: THE RING, Lies between them on the table.]
GANDALF : No, Frodo. The spirit of Sauron has endured. His life force is bound to the ring and the ring survived. Sauron has returned. His Orcs have multiplied...his fortress of Barad- dur is rebuilt in the land of Mordor. Sauron needs Only this ring to cover all the lands in the second darkness. He is seeking it, seeking it, all his thought is bent on it. For the ring yearns, above all else, to return to the hand of its master: they are one, the Ring and the dark lord. Frodo, he must never find out.
So, these two taken as a whole tell us that Sauron not only poured his own qualities into the ring during its construction, but he actually tied it into his OWN power & life-force. Therefore, whatever measure of power & evil we equate with Sauron, we must also equate that with the Ring itself, since it is essentially an extension of Sauron himself. The "how" exactly is never explained in the films, but they definitely explain the question of "why" the ring is considered so powerful and evil.