In American slang what does the word Jasper or Jaspers mean?

First used as slang in 1896 (according to the Online Etymology Dictionary), to call someone a Jasper is to call them a rustic simpleton... in other words, a hick.

Example: Did you see that dumb jasper in those dirty overalls?

In more modern times, this use of "jasper" has mostly been supplanted by the word, "bubba", another name common to rural areas.

In rural communities, it loses the insulting quality, and is used to mean "guy" or "man".

Example: Doesn't matter how cheap it is if some jasper doesn't want to buy it.

As with most community-style slang (like the use of the "n-word" by African-Americans), it's best not to use it if you're not part of the community.