Above is a dot I drew. Yay! As I make clear above, a dot is a type of point. Formally speaking, however, a point in space is less a visual concept and really more of an abstract, geometric notion. Euclid began his Elements with the following definition of a point:

Therefore, geometrically speaking, points have zero dimension, area, borders, or boundaries... all that defines them are their positions in space (thanks to Descartes). In the tangible world, however, one can make a dot on a piece of paper, as in my example at the top. Though this dot has slight dimensions, we really only interpret it as a point because of the principle of scale (which we’ll get to…). Increase the scale of the point, and you can see it is really a shape or form (and an odd one at that):

The Vanishing Point

Not all points used by artists and designers are visible. In fact, I'd wager that most are not. For example, the vanishing point – a staple of linear perspective drawing since at least the time of Brunelleschi (read more about that here) – is an abstract "point at which receding parallel lines viewed in perspective appear to converge (OED)." We can't see the vanishing points in a composition made with perspective in mind, but we know they're there because the angles of the elements in the composition tend towards them. 

Focal Points

Here's where it gets a little tricky. A focal point is also a type of point, one that our eyes are drawn to for a number of reasons. Contrasts of scale, apparent mass, color, positioning, and other attributes can imply various focal points in a composition (see here for a well-known example). This means that any element, regardless of scale, can be a point as well. Claude Monet's Impression, soleil levant (1872) makes great use of bright orange to create something slightly larger than a dot to create that painting's focal point. 

Below are some strategies for creating focal points.

Positioning (The Rule of Thirds)

The rational part of our brain might tell us that if you want an object or element to be the focus of a viewer's attention, just plop it right in the center of the frame. However, our instinctive brain doesn't work that way, and it turns out that, due to the way our brains instinctively scan our visual field for information, putting something right in the center of a composition makes it static: no motion, no visual interest...i.e. a lack of focus. 

In the images above, a portrait and a landscape format are trisected by two vertical and two horizontal, blue lines – into thirds. Where those blue lines intersect (circled in red), you will find natural focal point locations. Called The Rule of Thirds, Photographers use these focal points all the time to create visual interest and audience focus in their work. This isn't a hard and fast rule, and there isn't solid scientific evidence that these third zones are better than any other position, but there's no denying that the uneven spacing they enforce certainly helps content flow. Let's demonstrate with a picture of a sweet little girl!

On the left is the original photograph I took of my daughter Sloane, with her face in the center of the frame. She's super-cute, I admit. But the composition is not particularly interesting. On the right, I have placed our third lines, their intersections highlighted in red again. You can see that there isn't anything notable within those zones, and this is perhaps why this picture is kind of a yawn fest.

I cropped out some of the less interesting bits of the above original photograph and, seeing as faces are common focal points in any portrait, I make sure to place Sloane's cute little face within the top lefthand third zone. This composition is unambiguously about her, looking at toys, doing cute stuff. In other words, simply by cropping and positioning to a focal point and content flow, this composition now tells a little story. As Reinhardt famously said, "if it isn't worth a thousand words, the hell with it."

Scale, color, space, and the focal point

This may seem obvious, but another way to create a focal point is with the contrast of scale and/or color. In the image above, taken from Kate Greenaway's 1886 children's illustrated nursery rhyme book A Apple Pie, the large uppercase A is the clear focal point, not only due to scale, but also due to its relatively saturated fill color and black outline. Also used here to great effect is negative space, which omits any background objects that might distract from the composition's main focus: The letter A, the words "Apple Pie," and the children dancing around a massive pie. Scale, color, and space are all used here to create a hierarchy of focal points.