Don't let the terminology trick you. The term "small aperture" refers to the physical size of the lens opening. This may seem counterintuitive since the actual fstop number is larger. Yet, f22 is considered a smaller aperture than f16 because the aperture blades inside the lens don't open as wide. See the example below.
When you hear photographers say they are "stopping down", it means they are using a narrower aperture opening, for example from f8 to f11, or f11 to f16. This renders everything sharp from near-to-far, hence the phrase "great depth of field". I've created a phrase to help you remember this. "The greater the fstop number, the greater the depth of field."