Considering a home pc/laptop as machine used (Say typical 2.4 GHz, 16GB RAM, 4 core processor) for running any factorization algorithm. What would be the largest number that could be factored into its prime factors in milli seconds ? rather how many digits such number might have ?
We need a little bit of clarification or state some reasonable assumptions. For instance, 2^n can be factored very quickly for ludicrously large values of n. I assume that's not what you're asking. You're also probably not looking for a rigorous worst case, as that depends on the algorithms and implementations. Perhaps time taken for non-trivial random inputs.
"This can be yours for only pennies a day*" (*: approximately 64,000). By milliseconds, I'm assuming you mean something in the range 1-999.
On my computer, using Pari/GP or Perl/ntheory, looking at average time, 20 digit semiprimes take about 1ms, random input about 1/3 of that. Random input is about 10ms at 31 digits, semiprimes at 26-27 digits. Random input is about 100ms at 51 digits, semiprimes at about 38-39 digits. Random input hits 1000ms at ~60 digits.
Those programs are handy, but certainly not state of the art for factoring. Among other things, they only use 1 core. Something like yafu would probably be a better benchmark, as its SIQS is extremely fast and uses multiple cores. Last time I checked, the QS/NFS crossover was in the 100-110 digit range, and well over 1000 milliseconds.
As for algorithm, after simple things like trial division and maybe even a tiny bit of Rho/Brent, we're probably looking at ECM or QS. P-1 may be in there as well. SQUFOF is very nice for 64-bit inputs, but that's sub-millisecond timing for good implementations, and it doesn't work so well for larger inputs.
This is pretty old, but back in 2009 I did some tests on various factoring code. On page 18 of this presentation it shows the number of digits possible for factoring semiprimes in 1 second to be 55 digits with yafu. That was a Q6600, and yafu's latest code might be faster as well.