It really depends on what you are going to call a "computer."  The first meaning of the word applied to a person who could do complex calculations quickly, so in that case I think we'd have to get into some kind of evolution creation debate to answer the question.

If you consider it to be a mechanical invention, it is probably a contest between the abbacus and other simple computational devices.

The first computer we think of as being like computers we know, powered by electricity, etc., might be considered the Zuse Z3 invented in Germany in 1941.  We often point to the ENIAC which was operational in 1946, but it really depends on the perspective, etc.  The link below gives a great chart you can use to pick which one you think is the first.