This may vary somewhat from pet to pet, but in general, we're looking at two groups of test values: a "chemistry" and a "CBC".
A blood chemistry looks for chemicals in the blood produced and regulated by your pet's organs and glands; the quantities and proportions we find of each chemical can tell us important things about how your pet's organs are functionings. This, in turn, tells us if anesthesia or medication will be safe. It can also alert us to disease or unhealthy stress even when a pet is showing no symptoms at all.
CBC stands for Complete Blood Count; it involves looking at the blood and measuring the different types of cells in it (you probably know that blood contains red blood cells, but it contains several other kinds of cells too). These different sorts of cells should exist in certain proportions in a healthy animal; if there are more of some or less of others than there should be, this can indicate anemia, infection, or other serious problems.
If your pet is older or has known thyroid issues, we may want to check the thyroid as well - this could involve several tests, but the most common are a "T4", "Free T4", or "T3". These tests all look in different ways for hormones produced by the thyroid gland, which regulates many of the body's basic functions. Because the thyroid is so important, any problems with it are potentially fatal and must be treated. If we're treating with drugs, we need to make sure those drugs are being effective.