The white rose is the symbol of the House of York.
On the other hand the House of Lancaster's symbol is a red rose.
Henry VI was in the House of York while Edward IV was in the House of Lancaster. The two houses competed to take the throne and when one died the heir would take over.
Eventually it came down to the two princes of Lancaster and Richard III. Richard locked them up in the tower of London and ended up killing them.
Richard III took the throne after the two princes "disappeared". When it was time Henry VII battled Richard III at the Battle of Bosworth.
Richard died at that battle making Henry King. As king he married Edward's daughter and joined the two houses.
Henry VII founded the House of Tudor. The House of Tudor is the House of York and the House of Lancaster joined together. Hence the red and white rose symbol.
Credits:
Created with images by Dorothy Cook 15 - "rose" • Ravilbaptist - "white rose love" • Christopher Kray Visuals - "Red Rose Macro" • Josch13 - "red rose rose rose bloom" • Jangra Works - "Charming Valentine's Red Rose Flower #3 (Happy Rose Day)" • Josch13 - "red rose rose rose bloom" • Josch13 - "red rose rose rose bloom" • Hans - "apple blossom apple tree blossom" • _zhang - "Will you?" • MichaelGaida - "roses flower blossom" • p_a_h - "Richard III" • andybullock77 - "The last king of England slain in battle" • zaphad1 - "richard 3rd threshold" • pipilongstockings - "War of the roses RIP pops x" • hansbenn - "flower blossom bloom"