In the case of a simple pendulum (also called a mathematical pendulum of simple gravity pendulum), one assumes that all of the mass is the bob and the rest of the pendulum is massless. An example of the simple pendulum is given in the image below.
The simple pendulum (see wikipedia or hyperphysics) leads to a simple differential equation by using Newton's second law:
This pendulum gives the easiest way te look at harmonic motion. The above case is what they call the simple pendulum.
You could add an extrernal source so it would be a driven simple pendulum, of friction so that it becomes a simple pendulum with friction.
If you want to further complexify it, you could drop the assumption that all of the mass is in the bob and add inertia to the picture (so a real-life pendulum). This kind of pendulum is called the physical pendulum (or compound pendulum), the swinging body is no longer considered a point mass, but a mass with finite measurements. An example (compared to the simple pendulum) is given on the figure below.
This also leads to equations using Newton's second law applied on angular momentum (just as the ones which were used to derive the equation for the simple pendulum).