How many calculations per second can a computer make?

This depends on the speed of the processor. For example, a 2GHz processor can do 2 000 000 000 processes per second.

It isn't as simple as this however, since most processors have math co-processors within them. This is a less power processor which concentrates solely on calculations.

Note that I said 2 000 000 000 processes and not calculations. This is because processes do far more than just calculations. They are involved with putting data into memory and retrieving data from memory when required.

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Allow me to improve, if I could.

a 2.0GHz Processor does 2,000,000,000 CYCLES per second, not processes. The vast majority of processors do more than 1 "process" or instruction per cycle. For example, the Ivy Bridge i7 processors. They do a whooping 3.5 Instructions per cycle. This means that, if the processor was set to 2.0GHz, it would be capable of the following

3.5IPC*2.0GHz = 7,000,000,000 Instructions per Second

Another more common measurement is the FLOPS measurement, or Floating Point Operations per Second. An i7 3770k is capable of roughly 150GFLOPS, or 150,000,000,000 Floating Point Operations per Second. When somebody asks for a "number of calculations per second" count, this is the number they want typically.

Amazing stuff, isn't it?