The classic example is the halting problem: create an algorithm that takes as input a program in some Turing-complete language and some data to be fed to "that" program, and determines whether the program, operating on the input, will eventually stop or will continue forever. It is trivial to create an algorithm that can do this for "some" inputs, but impossible to do this in general. For any characteristic of the program's eventual output, it is impossible to determine whether this characteristic will hold.