Rifle scopes are available in almost any configuration of size and power imaginable. The most common scope by far is the 3x-9x with a one-inch tube, with varying sizes of objective (front) lenses. Some are available with an adjustable objective (AO) option, which can help correct for parallax at various ranges. But we're not going to deal with all of the technical aspects of scopes in this article - just the basics.
Why Scope it?
Hunters and target shooters use scopes for varied reasons. Some folks find that as they age, their eyes simply can't focus on iron sights like they used to, and going to a scope is a way to combat that effect. Others choose scopes because their hunting requires them to get a super-good look at their game, and the magnification offered by a scope allows them to make sure, for instance, that the seemingly-antlerless deer they're sighting on is not actually a small buck. Still others prefer scopes simply because they are so easy to use and often allow for more precise shot placement.
Not the Fastest
No doubt about it, a scope is not the fastest gun sight. That honor belongs to the peep (aperture) sight. Absolutely nothing beats a peep for speed and accuracy when hunting close cover, when identifying the game can easily be done with the naked eye. But the scope IS the easiest sight to use, because all one has to do with a properly zeroed scope is put the crosshairs on the sweet spot and squeeze the trigger - no sight alignment is necessary.
I hunted for many years without a scope, and I did well. But I found myself increasingly in circumstances that made me wish fervently for a scope. Using binoculars to examine a deer to confirm its legality is fine, but often by the time it's identified, there is little or no time to lower the binocs, raise the rifle, and take the shot. With a scope, it's fast and easy to make the shot once the game has been identified. These days I very rarely head to the woods to hunt deer (or squirrels, for that matter) without a scoped rifle.
Choosing a Power (magnification).
As I mentioned earlier, the most popular scope is the variable-power 3x-9x. Actual magnification values vary from scope to scope, but the numbers mean that at the lowest setting (3x), an object viewed will appear to be approximately three times the size it would appear when viewed by the naked eye - and at the top setting, it would appear to be about nine times that size. Adjustment between the low and high settings is infinite - you can turn the adjustment to any position between the low and high, and view the target at varying respective sizes. This is true of any variable-power scope.
For almost all hunting applications, 3x is plenty low. Even at a close ten-yard shot, you will be able to see plenty of your critter in the scope lens. I wouldn't want the low end of a hunting scope's adjustment to be any higher than 4x, because for close shots and/or shots at moving game, anything higher than that will narrow your view too much. And 9x is usually plenty high for zooming in on far game at reasonable ranges.
In some cases a higher magnification is desirable, but of course that depends on the type of terrain you hunt and how far the longest shot may be, and only you can determine your needs when choosing a scope. Anything above 12x is really overkill for most realistic hunting scenarios, and the higher you go with magnification, the more your every shake and tremble shows up in the movement of the crosshairs on your target.
My first hunting scope was a Weaver V3 1x-3x compact scope, which I mounted on my Ruger 44 mag carbine. This was a step up from the Williams peep sight I'd been using for years, but if I had it to do again I would not choose that scope - I would go with a higher-magnification model. If all my hunting was done in close cover, then it will work fine... but I often hunt in areas where distance to game may vary from right below me to 100+ yards out, and 3x is a bit wimpy for identifying game at longer ranges. A compact 2x-7x would probably be my choice if I were re-scope that gun today. The deer rifles that I use most often have 3x-9x scopes on them.