What algorithms are used in modern and good-quality random number generators?
In R, the default setting for random number generation are:
- For U(0,1), use the Mersenne-Twister algorithm
- For Guassian numbers use the numerical inversion of the standard normal distribution function.
You can easily check this, viz.
> RNGkind()  "Mersenne-Twister" "Inversion"
It is possible to change the default generator to other PRNGs, such as Super-Duper,Wichmann-Hill, Marsaglia-Multicarry or even a user-supplied PRNG. See the ?RNGkind for further details. I have never needed to change the default PRNG.
The Mersenne Twister is one I've come across and used before now.
The Xorshift PNG designed by George Marsaglia. Its period (2^128-1) is much shorter than the Mersenne-Twister but the algorithm is very simple to implement and lends itself to parallelization. Performs well on many-core architectures such as DSP chips and Nvidia's Tesla.
At http://prng.di.unimi.it/ you can find a shootout of several random number generators tested using TestU01, the modern test suite for pseudorandom number generators that replaced diehard and dieharder. You can pick and choose.