Euler and Gauss are (for now, at least) in my opinion the greatest mathematicians of the past two millenia (1000-1999, 2000-). Al-Khawarizmi takes the cake for the millennium before that. Then it's Euclid all the way back ;)
We know almost nothing about Euclid: we can figure out when he was active to within a century or two, and according to Pappus writing 500 years later some of his students/followers lived in Alexandria where Apollonius studied with them. That’s pretty much it for biographical details.
The earliest remaining editions of the Elements have no author mentioned, and our source that Euclid compiled it is a brief remark from Proclus 700 years later. Most of what is in the Elements was results from earlier, and it’s all but impossible to break down which bits were first done when or by whom. Most of what we can see today of the Elements or Euclid’s other books is later copies, much of it probably added/changed/reordered/... later.
If you want a 2000-year-old idol, go for Archimedes.
In the last 1000 years, the most influential mathematician is surely Newton, with an honorable mention for Leibniz. For the computer age (from 1950 through the upcoming few centuries), I’d put my vote on Grassmann (1809–1877), though his work was long ahead of its time and still substantially underappreciated.
Euler and Gauss were of course both brilliant and prolific and well worth studying, along with Descartes, Lagrange, Riemann, Poincaré, ....
https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem
(combining two of history's greatest mathematicians with names that have confused generations of students by being pronounced very differently)