Indeed, in true astronomers fashion, we classify everything. There are 3 classifications that stars can fall into, roughly based on their enrichment of heavier elements.
Population 3 stars correspond to the first generation of stars, which had to have formed without the presence of anything above Helium (I'm neglecting lithium because it was scarce). Most importantly, star formation is very different if there is no dust to form molecular hydrogen but I digress. Also, these stars have never been observed, they're properties are only theorized.
Pop 2 stars would consist of stars which formed from the nuclear processed material of the pop 3 stars. This can include everything from the hydrogen fusion processing to the rapid neutron capture heavy elements that you are familiar with.
Our sun falls in the pop 3 category, having been heavily enriched from many cycles of nuclear processing and is relatively young (and dirty).