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One of the earliest machines designed to assist people in calculations was the abacus which is still being used some 5000 years after its invention.
In 1642 Blaise Pascal (a famous French mathematician) invented an adding machine based on mechanical gears in which numbers were represented by the cogs on the wheels.
Englishman, Charles Babbage, invented in the 1830's a "Difference Engine" made out of brass and pewter rods and gears, and also designed a further device which he called an "Analytical Engine". His design contained the five key characteristics of modern computers:-
An input device
Storage for numbers waiting to be processed
A processor or number calculator
A unit to control the task and the sequence of its calculations
An output device
Augusta Ada Byron (later Countess of Lovelace) was an associate of Babbage who has become known as the first computer programmer.
An American, Herman Hollerith, developed (around 1890) the first electrically driven device. It utilised punched cards and metal rods which passed through the holes to close an electrical circuit and thus cause a counter to advance. This machine was able to complete the calculation of the 1890 U.S. census in 6 weeks compared with 7 1/2 years for the 1880 census which was manually counted.
In 1936 Howard Aiken of Harvard University convinced Thomas Watson of IBM to invest $1 million in the development of an electromechanical version of Babbage's analytical engine. The Harvard Mark 1 was completed in 1944 and was 8 feet high and 55 feet long.
At about the same time (the late 1930's) John Atanasoff of Iowa State University and his assistant Clifford Berry built the first digital computer that worked electronically, the ABC (Atanasoff-Berry Computer). This machine was basically a small calculator.
In 1943, as part of the British war effort, a series of vacuum tube based computers (named Colossus) were developed to crack German secret codes. The Colossus Mark 2 series (pictured) consisted of 2400 vacuum tubes.
Colossus Mark 2 (photo in public domain - copyright expired)
John Mauchly and J. Presper Eckert of the University of Pennsylvania developed these ideas further by proposing a huge machine consisting of 18,000 vacuum tubes. ENIAC (Electronic Numerical Integrator And Computer) was born in 1946. It was a huge machine with a huge power requirement and two major disadvantages. Maintenance was extremely difficult as the tubes broke down regularly and had to be replaced, and also there was a big problem with overheating. The most important limitation, however, was that every time a new task needed to be performed the machine need to be rewired. In other words programming was carried out with a soldering iron.
In the late 1940's John von Neumann (at the time a special consultant to the ENIAC team) developed the EDVAC (Electronic Discrete Variable Automatic Computer) which pioneered the "stored program concept". This allowed programs to be read into the computer and so gave birth to the age of general-purpose computers.