Euclid's proof proved that there was an everlasting set of prime numbers. He was the first person to prove this. It is a proof by contradiction — it assumes the opposite of what it is trying to prove and then shows that this is not possible.
The proof states the following:
If there is a limited set of primes, then all these primes can be multiplied together to attain an integer result.
However, if you add one to this number, the resulting number will be one more than a multiple of all the primes in the set and will therefore not be divisible by any other prime factors. Therefore, this number is a prime outside of the original set of primes.
This proves that you cannot have an finite set of primes and that there are infinitely many primes.