Unlike many other plants, the scientific name for these flowers is the same as one of the common names. Impatiens wallerana or hawkeri are often simply called "impatiens." This is true for every cultivar.
"Busy Lizzie" is another common name for impatiens. This name references the fact that the flowers spread quickly and have very long-lasting blooms. Indeed, with proper care--impatiens need consistently moist, rich soil--they will "busily" produce flowers all summer long.
Impatiens do not tolerate foot traffic. In fact, if you touch one that has a ripe seed pod, the pod will break open. This is why another common name for impatiens is "Touch-Me-Not." In this way, the flowers do reseed themselves, but often the seedlings will not look like the original plant.
Impatiens were first discovered, and the seeds collected, by a missionary in Zanzibar. The flowers were initially given the name of I. sultana to honor the Sultan of Zanzibar. Later, the name of the species was changed to Impatiens wallerana. This was done to honor the missionary who originally collected the seeds, Horace Waller. Still, some people today refer to the flowers as "sultanas."
"New Guinea" is the common name for Impatiens hawkeri, which are easily identifiable by their distinctive, purple, gold or bronze leaves. The flowers are large and showy. These plants tolerate sun better than Impatiens wallerana, but need constant watering. For this reason, they grow best if protected from the direct rays of the afternoon sun.