Several things...
First, a running back is not "supposed" to run through contact, nor is he "supposed" to avoid contact. If a hit is imminent, then yes, the running back is taught to lower his shoulder and drive through the hit. However, if we're just talking about "proper" playing style, it all depends upon A) the play calling, and B) the running back. A smaller, faster running back is going to be more adept at evasion, especially during off tackle rushes, sweeps, reverses, etc. A larger, slower running back is not going to be able to effectively run side-to-side, and thus running through contact up the middle or off guard is more productive for that style of back. Neither style is right or wrong.
Second, as a Tennessee fan, I have to be an asshole and post this in response to the Ben Tate video:
Third, Tate's speed and strength legitimately knocked Eric Berry on his ass, but I honestly don't see where he lowered his shoulder. I'm not attempting to call Tate's talents into question, nor am I simply supporting a Bama fan just to get everyone's panties in a wad, but I just don't see it. This video shows the hit from behind Tate, and I don't see any vertical change in the position of his shoulder. Additionally, it doesn't even look like he intentionally attempted to run through Berry; he was sidestepping the first tackle and happened to hit Berry. Props to Tate for having the ability to withstand the hit and knock Berry on his ass, but it just doesn't look intentional to me.
But, if you still have a hankering to see a running back who does lower his shoulder and really trucks a Tennessee defensive player, then enjoy this classic: