USA is republic because they are not monarchy or commonwealth, unlike Canada and Australia - they are commonwealth under UK.
USA is completely independence since 1776 and their government is run by republic - not relative to Republican Party (they are party with different philosophies).
There is similar between US and Canada - federal representatives are run by district and the district has to redrawn over time when population changes. They are running different system - congress and parliament. Canada doesn't have gerrymandering issue because they are redrawn by independent agencies, but in US, gerrymandering is big issue because most redrawn are done by state legislature - that why we are stuck with most of incumbent politicians or same parties for years. If gerrymandering isn't issue so we will see major shift between R and D over years.
The parties have been evolved a lot and Democratic Party was conservative in 19th Century, but most of them are left nowadays (major change was in 1960s - known as progressive). Republican Party is split into several classes, such as faith (progressive - started in 1970s) and business (classic/traditional/libertarian).
Interesting, the conservative in UK and Canada is based on toryism - they are middle of R and D.
http://www.thewire.com/politics/2013/08/how-conservative-would-uk-conservatives-be-us/67930/