What was the world's first computer?
The credit for the first programmable digital computer goes to
Colossus which was an Allied WWII code-breaking machine at
Bletchley Park. For other uses of the term computer besides the
modern "digital computer" one, an earlier answer might apply. For
additional information including a discussion of analog and
mechanical computers, non-programmable digital computers, and the
general history of machine computation, see the Wikipedia article
on "Computer".