"Sharpness" results from two major categories: technique and equipment.
Technique
Technique is by far the major culprit in the sharpness issues I've seen. As others here have suggested, when dealing with motion in the composition (e.g., animals moving, wind blowing vegetation, people moving, etc.), fast shutter speed is critical. Many wildlife photographers aim for 1/500 or faster for sharp wildlife shots, for example. This often means compromising depth of field (with a wider aperture/lower f-stop) and/or higher ISO (risking additional noise).
Another technique consideration is how the camera is held in position. There are many times you'll want a sharp shot that must be taken handheld and this is where one must learn and apply good stabilizing technique. There are plenty of videos and other resources online to demonstrate the concept, but learning how to brace your arms in a good support structure for your camera is essential. As important as handheld technique is, remember that this is like VR/VC/IS technology: It helps reduce motion in the camera body/lens, but will not eliminate it (pointing back to the need for higher shutter speeds).
Under the banner of technique, learn your lens' sharpest aperture range and use it when you can. Every lens has a sweet spot for sharpness and, while each lens has its own, most lenses seem to be sharpest at f/8 or f/11. Opening the aperture wide open (say, f/4 or f/2.8) and closing it to its smallest value (say, f/22) can result in noticeable loss of sharpness (and I'm talking about sharpness in the plane of focus, not the blurred fore-/background). There are lots of resources online to describe the physics behind this phenomenon.
Equipment
Even with absolutely perfect technique and even placing the camera on a tripod, you may find that your images still show some softness. One good first step is fine-tuning the focus (as mentioned by others in this thread). Not all cameras support this, but it's a good investment of several minutes of your life if you're suffering from soft images with a tripod-mounted camera.
Speaking of tripods, if you have one, it must be a solid tripod. I had a friend recently buy a tripod from Walmart ($30) and no matter how tightly she adjusted the various platform controls, there remained significant movement in the camera position. Tripod investment is truly a "buy once, cry once" scenario.
Finally, lenses have different levels of quality. "Kit" lenses are generally not as sharp as more expensive lenses - but there is not a direct link between price and sharpness. I have taken the same shot at the same settings with my Nikon 70-200 f/2.8G and my 28-300 and the difference in sharpness is obvious and rather striking. Lens acquisition is another area of careful consideration and is most often a result of balancing budget constraints with desired quality.
Oh, one other "equipment" consideration is the possibility of a malfunctioning lens. Don't drop them and you'll likely avoid this particular symptom.