One cannot discuss focus without first explaining Depth Of Field. Any given lens (focal length), set to any given aperture (stop), and focused on any given subject that is closer than "infinity" (that figure-8 marking on the barrel), will have an area both in front and behind the subject that is in "apparent" focus. This area is called Depth Of Field. It increses as focal length and aperture decrease, and as subject distance (to camera) increases. Also, the subject in perfect focus (plane of focus) is about one third back in the depth of field (area of apparent focus).
If you are curious about the finer points of focus theory ("depth of focus" as opposed to "depth of field", hyperfocal distance, circles of confusion, etc.), I encourage you to read "The Camera Assistant" by Douglas C. Hart. It is an excellent book and is available at most specialty arts and film/theatre bookshops, so please don't ask me to get into these mathematical complexities on this page.
What I do want to talk about is aesthetic focus choices. Having a very small depth of field, or, conversely, keeping your entire frame acceptably sharp, makes a huge impact on the mood and feel of your shot, as well as affecting how the story is told. Deciding what parts of the frame are in focus and what areas are obscured is as important as your lighting choices, considering all elements of composition and design; focus is not only a visual, but a narrative element as well.
Following focus, or "pulling", is not easy, and it takes skill and practice to keep your subject sharp if either it or the camera, or both, are moving. In a situation (let's say we're shooting 16mm) where the shot calls for a 10mm lens and the stop is set at T8, and the subject in the scene is walking away from camera, a professional 1st assistant (commonly called the "focus puller") will likely consider this a good time for a coffee break. However, if the following shot calls for a close-up of the subject walking around the next set, using an 85mm at T2, you'll have a very nervous and irritable focus puller to deal with. (Especially if your actors never manage to hit their marks, or even know what the heck those colourful strips of tape on the floor are actually for... and this is all too often the case.)
Pulling focus by simply gripping the lens barrel is generally not a good idea, and should be avoided as it leads to a likelihood of sloppy/jumpy transitions, missing the barrel mark, or worse, a finger in the frame. A "follow focus" (see Accessories) is an apparatus that will give you more leverage, spreading the distance from focus point to focus point on the lens over a greater distance, and thus helping you achieve smoother transitions. If you don't have a follow focus, many lenses have wee holes which specially-made rods screw into (or you can assemble a homemade jobby with tape, a pencil, whatever) for that extra leverage and control we all need in our lives...
The most common lens tests are for sharpness, colour, and field curvature. More on this another time.