Hey Mex, welcome to the best hobby in the world. I'm not a major expert and I'm sure others will chime in, but I'll give my best explanation as I see it. Military plates generally consist of three categories-- the belt buckle, the ammo cartridge box plate, and any plate worn above the belt line like the Union army ones affixed to the knapsack strap. They are basically all of the styles and more that you see in this thread, a significant piece of flat metal that usually is adorned with some type of military decoration. Although there are plenty of civilian decorations as well, we focus on the military ones in this thread.
I'm not sure, without looking it up, when plates were first made, but I believe it was sometime during the 1700's. The ones found here in America are anywhere from the 1700's, 1800's, or even 1900's. Most found on this forum are from the Revolutionary war or the civil war.
Many union plates from the civil war have either an Eagle, the letters US, or a State seal on them. The confederate examples include CS, CSA, or southern state seals or depictions. Many variations and examples exist though of different styles and depictions on the front for various groups, regiments, brigades, etc. Some plates have nothing on them at all.
As far as flat buttons, they are just that...flat, one piece buttons made from a singular piece of metal with a loop called a shank affixed to it. Again, no expert here but I believe they were made from early 1700's or even earlier, up into the mid 1800's. They are a common find around old house sites and many variations exist.
Hope this helps