What constitutes a Notice?

What is the name of your state? Texas

What if you NEVER gave a notice to your employer, but discussed leaving with another employee (not of the same company, but in the same place of employment), and HE told your employer you were leaving and gave the employer a date? Is that a notice? My UI was denied because the employer said I quit and gave notice to HIM. What constitutes a notice?

Pretty much what the employer wishes to consider as "notice." You won't find one exact legal definition, if that's what you're wondering.

In the circumstances you describe, it would be worthwhile to file for an appeal (i.e. a hearing) with the UC Division if it's not too late. I can't predict what they'll rule as I expect there are a few more facts than what you've included in your post however.

That's pretty much it. My former employer was told I was leaving by a third party, and he fired me saying he was "accepting my resignation". Is there a particular format an appeal should be in, and where can I find it? Or is it just a paper in my words?

Information on how to file an appeal will have been included with the Initial Determination you received. Don't delay as there is typically a very short period of time you can do so.