What is wrong with my car? It wont start!

The night my car stop cranking up I had just drove it home from work around 6pm. Around 825pm, I get ready to leave and my car doesn't not start. The things I have checked and ruled out so far is the battery (because I went and bought a brand new one), corrosion on the battery (used Coca-Cola) and loose battery cables. I change own oil a lot and my mother mentioned I had a leak but my oil dash indicator never lit up,my car never over heated, nor did I find any oil in the drip pan after placing it there for the day.  So, I didn't bother to check it. So, as of now, when I try to turn my car on all I got is a single click from the engine and a momentary buzzing noise which is probably suppose to indicate where the problem is.

I've had a minor problem that seemed to went away and that problem was that my electrical components in the car would flicker on and off and sometimes shut the engine down while I'm driving bringing the car to a slow stop. But right after that, my car starts up like nothing happened. This minor problem wasn't constant but it happened mostly during the winter time with snow and ice. This minor issue seemed like nothing.

But the most important information I can give is that I was not driving when the car decided to not start anymore and I just used the car 2 hour prior to the discovery of my current issue.

What was this minor problem? Did it lead to my engine not starting at all? Why isn't my engine not starting? Motor locked up? Seized engine? Could it be the Starter or Alternator? Can I save my engine!?! Cost estimate?