Eukaryotic cells are those that make up multi-cellular organisms such as plant and animals. They have separate organelles, each with a specific job to do in order for the cell to function properly. Over the course of the cell's life span, it takes in and disposes of nutrients. New organelles are formed, and older ones are broken down and disposed of. It is the job of the lysosome, a membrane-bound organelle which contains various enzymes, to break down materials which the cell needs to get rid of.
More specifically, the lysosome is related to the nucleus, as the directions for constructing the lysosomes are contained in the DNA. They are related to the Golgi apparatus in that they form by "budding" off the Golgi by pinching off a bit of membrane, containing the digestive enzymes.