A video tripod definitely helps, but sometimes you have to shoot handheld. These tips will help you shoot smooth, steady video when you don't have a support system to rely on.
Use Your Body
When I shoot handheld video, I try to use my body as a tripod by adopting a wide stance to give me a sturdy base and help keep the camera steady. Then, I bend my elbows and keep them tight to my sides. This position stabilizes my hands and body so they don't move the camera much. When I want to shoot from a lower angle, I hold the camcorder from above with one hand, and support it below with the other.
Don't Move the Camera
Moving the camera smoothly is very difficult when you're shooting handheld video. Whether you're trying to walk, pan or tilt, the timing and the motion will never be anywhere near as smooth as if you're shooting with a tripod or a dolly. Instead, it's much easier to keep the camera still and let the action play out in front of the lens.
Improv a Tripod
Really, anything that can support your camcorder can serve as a tripod. If there's a wide, flat surface you can rest your camera on that. Or, rest your elbows on something steady as you hold the camera in your hands.
Avoid the Zoom
Zooming in with your camcorder lens will exaggerate any kind of jitters in your footage. Instead, keep the lens as wide as you can, so that the shakiness is less evident. If you need to get a closer view of something, move physically closer instead of using the zoom.
Practice, Practice, Practice
As with many other things in life, you'll get better at handheld videography if you devote some time to practicing. Hold the camera in different positions to see what works best for you, and then practice holding that position. You can improve your muscles and your posture so that you can hold the camera steadier, for longer.
Also, practice walking while holding the camera, and practice making moves like tilts and pans. As I noted above, movements are difficult to do well without a tripod, but you can get much better at them through practice.
Embrace the Handheld Ethos
In general, these tips are directed at people who are shooting video without a tripod yet still want footage that is smooth and steady. But that's not the look that everyone is going for. Many directors - from the makers of The Blair Witch Project to reality show producers - deliberately use shaky camera techniques in their work, and you can too.
Most professional and consumer camcorders have an image stabilization feature built in. Turning this on can make a huge difference in reducing the shakiness of your shots. If you don't know if your camera features image stabilization, check the manual.
Fix It in Post
Thankfully, video editing software can also do a lot to fix shaky handheld footage. Even free editing software comes with image stabilization features. And Apple's iMovie makes it a built-in option for any footage you import.
Of course, just because you can fix a lot of this shakiness during editing, doesn't mean that you should rely on it entirely. Instead, it should be a back up after you've adopted these other strategies for getting steady footage during shooting.