When you realise something, it becomes clear to you. Brits spell this word with an "s," while Americans will replace the "s" with a "z." We hope you realise this difference if you are editing a Brit's writing.
All the meanings of realise amount to making something “real” in some way. If you realise a plan, you make it happen. If you realise a fact, you see its truth. If you realise the seriousness of a situation, you become aware of its dangers. Since this word was originally borrowed from the French réaliser (meaning "make real"), we can understand why the Brits spell it r-e-a-l-i-s-e with an "s."