Three

Properties

3 is the first odd prime number, as it's only divisible by 1 and 3. 3 is a Mersenne Prime: (2^2)-1 = 3. Mersenne primes must be of the form (2^p)-1 where p is prime. This also means that 3 factors into a perfect number: 6, as all even perfect numbers are of the form M * 2^(p-1) where M is a mersenne prime.
Three is a triangular number, for obvious reasons.
3 roughly approximates to e, and less roughly into π.
Three points that aren't collinear can determine a plane or circle.
And of course, there are three spatial dimensions.

For different bases:

1. In Binary, 3 is 11.
2. In Ternary, 3 is 10. Ternary is considered the most "efficient" base, as it's radix economy is closest to e.
3. Quaternary, Septimal, decimal or any other base greater than a multiple of 3 has a divisibility trick: if a number's digits add to a multiple of 3, the original number is divisible by 3. This works if you substitute 3 for another number N, and the bases with numbers one greater than a multiple of N. For 3, this means most bases have an easy divisibility trick for it, including decimal and even hex, which also work for 9 and 5 respectively.
4. 6 is divisible by 3, so all numbers in seximal ending in 0 or 3 are threeven.
5. 9 is 3^2, so two ternary digits correspond to a nonary digit.
6. 12 is divisible by 3, so all numbers in dozenal ending in 0, 3, 6, or 9 are threeven.
7. 27 is 3^3, so three ternary digits correspond to one digit in trinonary.
8. In Negabinary, 3 is 111
   and -3 is 1101.
9. In Negaternary, 3 is 120
   and -3 is 10.