The "Rule of Thirds" is one of the most basic concepts in composing your shot.  By splitting the image with two lines vertically, and two lines horizontally, the intersections of these lines tell you where the subject of your shot should be.  Obviously, there are notable exceptions, but it's a "where do I start?" rule-of-thumb that helps a novice create visually pleasing images.  Splitting up the screen can be done a couple of ways: most high-end cameras (and smartphones) already can produce an overlay for it in the viewfinder, but if it doesn't, then it's time to construct a sort of "Tic-Tac-Toe" board to attach to the viewfinder to visualize the screen in 9 equal parts.

Reel Rebel: How to Use the Rule of Thirds

Here's Stephen Schweickart talking about the rule of thirds in another episode of Reel Rebel:

The rule of thirds was first written by John Thomas Smith in 1797 in his book, Remarks on Rural Scenery.  It took a concept from Sir Joshua Reynolds, who discussed the balance of light and dark in a painting.  Smith took that concept even further, coining the term "rule of thirds," for how the subjects in a painting should be separated.

For video, this is what the grid looks like:

rule of thirds

Your subject should be at the intersecting points.  For something like a horizon, that should lay right on one of the horizontal lines. It’s a really simple rule to follow. Just put your talent’s eyes on the cross you’re on your way to a halfway decent looking shot.

As always, once you're comfortable with the basics, breaking the rule of thirds can be used to create shots that you want to achieve a different effect.  Maybe you don't want to have a shot that's simply "visually pleasing," you want something that causes the viewer to be unnerved.  The rule of thirds is something that helps you when you just don't have any idea where to begin.