What is the relationship between the second law of thermodynamics and entropy?

Entropy means, roughly, "disorder." The second law says that, in an "isolated system" (that is, in any group of objects that does not have any added to it or any done on it), total entropy increases over time. Your room is a system, with lots of objects. Over time, your room tends to get messy (disordered). It never gets more ordered, unless you expend energy and do work to straighten things up. In other words, order tends not to happen spontaneously (without the addition of energy or work).

The growth of plants and animals -- building ordered structures from small building blocks (mainly molecules of water and carbon dioxide) is possible only because energy from the sun is constantly added to the Earth. Earth is not an "isolated system." The sun's energy permits plants, in photosynthesis, to make sugars which in turn become even more complex molecules and structures (fibers, wood, branches, etc.). Food energy from plants (or other animals that ate plants) allows animals to do the same.

Entropy does not just involve objects becoming more disordered. We also know that entropy increases when heat flows from relatively hot to relatively cold objects. The flow of heat from hotter to colder objects means that, in an isolated system, things eventually all get to be the same temperature. Hot cookies cool off. Water in the dog's dish outside freezes on a very cold night.

A smart student might ask: how could water, in a disordered liquid form, spontaneously freeze into highly ordered ice crystals? No energy was added. The answer is that when water freezes, heat is given off, which flows to the air surrounding the ice. This flow of heat increases the entropy of the air, so that even though the ice has less entropy than the water, the total entropy of the system increases.