A pseudo random process is a process that appears to be random but actually it is not. Pseudo random sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process. Such a process is easier to produce than a genuinely random one, and has the benefit that it can be used again and again to produce exactly the same numbers and they are useful for testing and fixing software.
The generation of random numbers has many uses (mostly in Statistics, for Random Sampling, and Simulation, Computer Modeling, Markov Chains, Experimental Design). Before modern computing, researchers requiring random numbers would either generate them through various means like coin, dice, cards, roulette wheels, card shuffling etc or use existing random number tables.
A pseudo random variable is a variable which is created by a deterministic procedure (often a computer program or subroutine is used) which (generally) takes random bits as input. The pseudo random string will typically be longer than the original random string, but less random (less entropic, in the information theory sense). This can be useful for randomized algorithms.
Pseudo random numbers are computer generated random numbers and they are not truly random because there is an inherent pattern in any sequence of pseudo numbers.