Some common examples of Newtonian fluids are water and air. While no fluid is perfectly Newtonian, most are close enough to be considered Newtonian. There are specific equations and calculations in the field of material science for both Newtonian and non-Newtonian fluids.
The key characteristic of a Newtonian fluid is a constant viscosity at any range of stresses. The viscosity can be thought of as the thickness of a fluid, or the resistance to flow it possesses. A non-Newtonian fluid is just the opposite, it can have a drastically different viscosity at different stress levels. A common example is whipped cream; the longer it is mixed, the thicker and more solid it becomes.