A set (e.g. "all numbers with a 5 in them") is said to be "decidable" if I can write a program (usually for a Turing Machine) to determine whether a number is in the set and the program will always terminate with an answer YES or NO after a finite number of steps.

Most sets you can describe easily are decidable, but there are infinitely many sets so most sets are undecidable, assuming any finite limit on the size (number of instructions or number of states) of our programs.

I.e. how ever big you allow your program to be there will always be sets which need a bigger program to decide membership.

One example of an undecidable set comes from the halting problem.

It turns out that you can encode every program as a number: encode every symbol in the program as a number (001, 002, ...) and then string all the symbol codes together.

Then you can create an undecidable set by defining it as the set of all numbers that represent a program that terminates in a finite number of steps.

A set can also be "semi-decidable" - there is an algorithm that is guaranteed to return YES if the number is in the set, but if the number is not in the set, it may either return NO or run for ever.

The halting problem's set described above is semi-decidable. You decode the given number and run the resulting program.

If it terminates the answer is YES.

If it never terminates, then neither will the decision algorithm.