All of technology uses mathematics, whether it is a car, radio, rocket or toaster. As @hiphiphopflipflapflop pointed out, the physicist Eugene Wigner spoke of the unreasonable effectiveness of mathematics. Mathematics is more than just numbers, but think of all the things that can be measured using numbers. We use them to count, find probabilities, and measure length, time, mass and electrical charge. Why should numbers be applicable to all these things?
My vote for most useful branch of math would be analysis, which includes calculus. I don’t think there is a branch of mathematics that does not have some practical use. The mathematician George Hardy took pride in saying that his area of expertise, number theory (the study of whole numbers), had no practical application. He has been proven wrong on this. There are several uses, notably public key encryption.
Apart from its practical uses, mathematics can stand on its own and, for those of us who appreciate it, can be quite beautiful.