Pick a point on your computer monitor, maybe the period at the end of this paragraph, and stare at it. Now wiggle your head around a bit, bounce up and down in your seat, or even try balancing on one foot while still staring at that point. What you'll find is your eyes and your brain start communicating and working together so that you can keep pretty good focus on that point - the human body really is an intelligent thing. Video cameras however, are pretty dumb in comparison. Luckily you can "smarten" them up with the use of a simple tripod.
A tripod is a three legged stand for your video camera. Why three? Well, four legged stands (tables, chairs, etc.) are great when you're on perfectly even terrain - but chances are you won't be - and when's the last time you saw an inanimate object stand reliably on two legs? With three legs, you can set up your tripod just about anywhere - even the side of a rocky hill - and keep it standing upright and sturdy.
The best part about tripods is not only can they really take your films from looking amateur to pro, but they can be fairly inexpensive too. Looking around online I found models as cheap as $60-$70, and if you aren't wanting to invest too much money into your equipment, three legs are three legs! That being said, if you plan on doing any smooth panning/tilting shots, you might have some troubles with the stubborn plastic mechanisms (although here's a neat solution).
Gorillapod line - still small enough to take on a vacation, but their flexible legs allow them to stand, grip and cling pretty much anywhere.
What more convincing do you need? Go get yourself a tripod - your viewers will thank you!