The best way for me to describe the differences between a phase, project life cycle and the process groups the way that I understand it is that the project life cycle is the entire project from cradle to grave and inside that life cycle lives various phases which must be completed within that life cycle to deliver a finished product. Deeper within those phases live the process groups which guide the phases to a (hopefully successful) completion. Not every phase uses every process group but the process groups can be repeated during a phase if necessary.
I’m having a struggle coming up with a real business/project related example of each but the best way I can describe it would be to liken it to making a pizza. The project life cycle would start with the raw ingredients, move to combining them, baking the pizza and finish with serving the pizza. Each step of making the pizza is a different phase: crust, sauce, cheese, pepperoni, bake, and serve. And then each one of those phases would require a process to create the phase. For example, the crust would involve creating dough, kneading the dough, letting it rise, and rolling it out. This is just of course my way of simplifying these concepts – please feel free to chime in and elaborate or if I don’t quite have the concept, fill me in.