Let me start with Wikipedia figure:
Above we have a wide open aperture. Only point 2 is in focus. Points 1 and 3 are out of focus. Due to wide aperture, the rays coming from them through different parts of the lens intersect the screen (film, sensor) in different points. Effectively, an out-of-focus point produces a circle of confusion. This is what we can call blur or bokeh.
For smaller aperture below, the rays too far from the center are cut off, so the circle of out-of-focus point is smaller. If they are smaller than film grain or sensor subpixel, we cannot tell they are out of focus at all, and then all points 1, 2, 3 appear as in focus. This may be called a deeper field of focus due to smaller aperture.
If the aperture is really, really small, then only central rays can pass, and we have infinite depth of field no matter what. Every point, close or far away, is represented as a point. This is how pinhole camera works. Adjustable aperture allows to have anything in between.