The Tudor Rose was ushered into the decorative and architectural stylings of England and Wales with the coming of Henry Tudor (King Henry VII) to the English throne.
Wresting the throne from Richard III through a victory at the Battle of Bosworth Field, which left Richard dead and rather famously sans horse, the Welsh Henry united the Houses of York (the white rose) and Lancaster (the red rose) via his marriage to Elizabeth of York in 1486.
The combined houses led to a combination of their respective symbolic flora, Henry VII taking for his own a white rose embedded within a red one. Typically, both roses have five petals, each slightly upturned at the end. The red, the larger of the two, it is usually oriented as would be an upright, five-pointed star, with each point ending up in the center of a petal. The white rose would fall along the pattern of an inverted five-point star.
The red rose is the larger because Henry Tudor was a descendant of the House of Lancaster, and he incorporated the House of York into his family line.
When I first wrote this, I didn't make a link to the War of the Roses. This may be because I'm a bit thick.