Am 17 and around December of 2011. I was heartbroken due to my ex and was a huge misunderstanding were the college and city police department got involve due to in older lady false accusing me! Of dropping her and kicking her (completely false). I was not pressed charges of course because it did not happen at all. But I was handcuffed by both the city and college police department but was let go. My question is whether this will show up on my background check for a best buy job that I applied for?

No, a Best Buy check will only look for convictions.