The abacus is a counting machine and an aid to human calculation. In itself, it can't compute anything.
The first genuine computer was invented by Charles Babbage in Manchester, Great Britain. It was a mechanical device. He started his collosal 'Difference Engine' in 1822. It had 25,000 parts and weighed 15 tons! It was never finished he realised how to improve it before he finished. But it was later completed by enthusiasts and proved to work. Babbage's ideas started research that leads directly to the early modern computers. He is accepted as having invented the computer.
http://en.wikipedia.org/wiki/Charles_babbage
The invention of the first modern electronic computer is hugely contested and remains unclear because 'a computer' is defined in different ways and everyone wants the 'glory' of being the first. It was probably The Germans who got their first. However, there is a progression of machines. Here is a list of contenders.
http://en.wikipedia.org/wiki/Colossus_computer
Certainly the most exciting early computer and a contender for the claim of the world's first programmable digital computer was built in secret by the British in the WWII at Bletchley Park. Bletchley is where the Nazi's Enigma code was de-coded, using computers. The Bletchley computer made an unimaginable contribution to winning the war.
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