One way of thinking about computers is as sophisticated calculators. A calculator is just a machine for performing arithmetic; these have existed for thousands of years, for example in the form of the abacus.
The main distinguishing feature between "just" a calculator, and a computer, is the idea of "universality": a calculator has a set of operations (e.g. addition, multiplication, etc). Each button corresponds to a fixed operation. A computer is able to run a more complete set of algorithms by using the result of one operation to decide the next operation. Read the Wikipedia article on von Neumann architecture; see also Turing machines.
Calculation before the era of electronics was mostly, but not entirely manual: the ideas of computation machines, and even computers, existed long before electricity was common-place. Read the Wikipedia article on Charles Babbage.