Did the US join in the war between Georgia and Russia?
No, the United States did not get involved in the conflict between Georgia and Russia. This was a very brief episode of actual fighting that occured in 2008. No countries other than Georgia and Russia were involved in the fighting.
The conflict arose out of a much longer-standing conflict between the two countries over some areas that are claimed by Georgia but which Russia thinks should be independent. Georgia claimed the regions of South Ossetia and Abkhazia, but Russia argued for their autonomy. This is largely due to to the fact that Russia would like to weaken countries like Georgia and Ukraine that were once part of the Soviet Union. They do not want these countries to be able to become allied with Western Europe and the United States because they think this would be a threat to their dominance of the region.
The war did not last very long and there was never any real chance that the US was going to intervene in a conflict with a power as big as Russia unless it was really important to US interests.