A number chosen for some reason. When stochaistically or algorithmically computed, random numbers have a relatively flat distribution.

When selected by intelligent processes, random numbers tend to have a much more complex scattering; often numbers that have personal or cultural value.

Some common randoms are 1 3.14159265358979323846264338327950288419716939937510... 7, 23, 42, 666, 999...

cmell is perfectly correct in stating that random numbers are chosen (read: generated) for a reason. In computer programming, random numbers are extremely useful. In most hard sciences, all simulation programs will use some form of random numbers to help model uncertainty.

It is important to note that random numbers need not have a flat distribution (also called a uniform distribution). They can be generated according to any conceivable probability distribution function, according to the needs of the researcher/programmer. Some common distributions from which random numbers are generated are normal, normal with skew, Fisher's F, Student's t, and the famous chi squared.

Is a random number chosen (or generated) at all?

See A random real is irrational. There is an argument that a random real number is not part of any countable set. Now see Science is countable, - all numbers that can be specified are countable. Can you say a number has been chosen if the number cannot be specified? It will always be "No, that's not the number" and never "Yes, that's the number". Can it then be called a chosen or generated number?

Pseudo-random numbers are generated, granted. But, it seems, not pure random numbers.