If by "computer" you mean general purpose programmable digital computer then most likely the mechanical Analytical Engine in the 1840s, but Babbage was never able to build it.
If you mean any computer (including special purpose analog) then most likely the mechanical Antikythera mechanism found in a shipwreck off Greece in 1900. It is believed to have been made and used in the 2nd century BCE.
If you mean electronic computer (including special purpose) then most likely the Atanasoff-Berry Computer (ABC) designed in 1937.
If you mean electronic general purpose programmable digital computer then most likely ENIAC designed in 1943.
If you mean "computer" as originally defined in dictionaries, that was not designed it evolved as the first computer was a human computer.
There are lots of "first computers".
The first actual working computer was designed by Raytheon Electronics in the late 40's.
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charles babbages
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The original computer that the first Apple computer was eventually created from was designed and built by Steve Wosniak.
Jon von Neumann designed the first computer virus in 1949. His design was based on his theories about a self-reproducing computer program.
Charles Babbage was 33 years old when he designed his first computer but he didn't build
Prepare firing tables for artillery and naval gunfire.
Osborn-1, designed by Adam Osborn.
The first programmable computer, the Colossus computer, was designed and employed by British codebreakers to help read encryped German messages during WWII.