The trees became nationally iconic in a bid, many years ago, for Japan to distance itself from China (whose national symbol is the plum blossom). In hindsight, they probably should have picked something a bit more different, as the two are often confused by outsiders.
The distinctly white, mildly pink (when in full bloom) blossom which dominates much of Japan belongs to a tree known as 染井吉野 (somei yoshino). This is the "standard tree". There are around 100 other varieties, such as the low hanging, droopy branches of 枝垂れ桜 (shidarezakura). Intensely pink blossoms most likely belong to 関山桜 (kanzanzakura).
Anyway, someone asked me the other day about the different types of cherry trees seen in bloom. So, here's the answer they wanted:
Click to enlarge.