How do you make custom shapes? In a lot of tutorials the authors start out with a custom shape that they don't explain how to make. By custom shape I mean something that isn't based on one of the primitive shapes, so no extruding, etc. A good example of what I mean is something like https://www.youtube.com/watch?v=QY7iJmqsJf8 when he shows the bullet he made but didn't show how. I am also curious as to how to go about this with more complex shapes, like for example highly detailed models.

I apologize if this is a very easy to answer question, but all of the tutorials I can find talk about advanced automatic features and completely skip the actual modeling which is what interests me most right now. 

To clarify, I am NOT talking about just subdividing something.. I am asking how to make shapes from scratch like seemingly almost everyone is able to do in a minute or two.

Thank you for reading and I appreciate all responses because I have a lot to learn.
That's more or less how it's done. You model them, starting from a primitive. 

For example, to make a bullet you might start with a cylinder or circle (ShiftA>Mesh) and then extrude (E) and scale (S) to top ring repeatedly, finally merging it into a single vertex (AltM):




Note that you must be in Edit mode in order to edit the underlying geometry of an object (you can toggle edit mode quickly with Tab).

For more complex models, you might use a different primitive. For example, most round objects (e.g. bowl) can be started with a circle, while for a more free-form object you might use a plane or even a single vertex:





This is a rather sloppy example, especially as I could have used a circle as the starting mesh, but I think you get the idea.



There are many great tutorials on modeling in blender (look around on youtube etc.), here's a list of some good resources.

Another thing you can do is watch modeling time-lapses like this one.