A tripod is a wonderful piece of equipment for assuring camera steadiness, but only if it's used properly. There are still ways of getting camera movement, even with the most substantial tripod.
With the camera mounted on a tripod, maximum steadiness can be obtained by using a cable release, if the camera has provisions for one. Most compact film and digital cameras do not. The danger in not using a cable release is that when pressing down on the camera's shutter release, you're also pressing down on the camera, and this can move it.
With the camera mounted on a tripod and without a cable release, you can get much better camera steadiness than you could by hand-holding the camera. Just hold the camera steady with both hands as you squeeze the shutter release. It's possible to get fairly good results with this technique down to a shutter speed of about 1/4 second, but a cable release is still more reliable. For you SLR users, if you're using a lens with a focal length of 150mm or greater at a shutter speed of 1/30 or less, always use a cable release.
When using a cable release, the cable shaft should be bent almost double. This will eliminate the possibility of jerking the camera while pressing the plunger.
You can use the camera's self-timer and get the same degree of steadiness as that obtained with a cable release. This is a practical option in scenery photography or any time that it's not necessary to take a shot at a precise instant, such as at a sporting event. If the self-timer is used, be careful when setting it so you don't knock the camera out of line, altering the composition.
With the camera mounted on a tripod, the use of a cable release is a good starting point, but there are a few other factors that affect camera steadiness:
Tripod legs should be angled out to the leg stops, unless the legs can be locked in at some other angle. In some tight spots or on hillsides, this isn't always possible, so do the best you can under these circumstances.
If it's not necessary to extend the tripod to its maximum height, get as much elevation as possible with the tripod legs, then extend the centerpost because the post is more subject to motion than the legs are. Also, extend the thicker legs first, then the thinner, flimsier legs.
Make sure that all locking devices are secure so that there is no slippage. This includes legs, centerpost, and tilting head locks.
If the terrain is uneven, extend or shorten one or two tripod legs so the centerpost is approximately vertical. If the post is at too great an angle, the weight of the camera can pitch the tripod over.
For maximum steadiness, place the tripod so that one leg is extending forward and you are standing between the two rear legs. This is especially important if a long telephoto lens is being used. When you move away from the tripod, step straight back so you don't bump the tripod legs on either side of you.
No, it isn't necessary to use a tripod all the time, even all situations in which a tripod would be appropriate. But a tripod will help provide more flexibility, creativity, and sharper pictures than any other photo device.