For those of us who are not mathematicians, what would quantum computing practically mean? Compare current math capabilities of computers and quantum computing in layman's terms, please.
Shor's algorithm allows quick factorization of primes. RSA encryption, the main way that things are encrypted today, relies on the assumption that factoring large primes is computationally infeasible. So quantum computers would allow the NSA to read RSA-encrypted data.
What we do know is that there is a known algorithm to factor number in polynomial time on quantum computers and there is not a known algorithm to do so on classical computers. Assuming that factoring numbers is actually a hard problem on classical computers (unproven, but generally accepted), then quantum computers are more powerful than classical ones.
I understood this as, it's impossible to run some algorithms on a traditional computer. You can run Shor's algorithm on a Turing machine, albeit requiring a simulation of qbits. You can run it in javascript and in your browser if you like.
Since Moore's Law is giving us one more bit of decryption per unit time every 1.5 years, 4096 * 1.5 years is probably going to precede the heat death of the universe handily. But 6000-or-so years of Moore's Law does tax credulity.
PGP is a piece of software, and not an algorithm. I believe it uses RSA and AES by default, but I could be wrong :)
AFAIK RSA, DSA, Diffie-Hellman, and ECC are all known to be vulnerable to Shor's Algorithm (which, granted, do make up a lot of the crypto in common use), but there are lesser-used algorithms like McEliece and NTRU that are not vulnerable.
That said, quantum computers are still very far from breaking modern cryptography. We're playing around with a handful of qubits, but to break something like a 4096 bit RSA key would take trillions of quantum gates.