A barrel with a 1:7 twist is one full twist of the rifling in 7 inches of barrel.
As far as which twist rate is best, there's a long and short answer. Short answer - for 50 to 62 grains a 1:9 will work fine. Anything with a faster/tighter twist and you risk over stabilizing the lighter bullet. Some shorter .223 varmints bullets will even lose their jackets and disintegrate in the air with a 1:7 twist. The original AR-15 had a 1:14 twist, but most barrels these days will have 1:7-1:9.
Long answer is all about gyroscopic stabilization. When a bullet leaves the barrel it is traveling extremely fast, and slight variances in atmospheric conditions between the muzzle and the target will impart angular forces on the bullet, which will cause a change of orientation as it travels, and it will take a different and unpredictable path to the target. This results in extremely poor accuracy. This is where the need for gyroscopic stabilization comes into play.
When a bullet is spinning at a very fast rate, any angular forces imparted on it will be resisted, and it will attempt to remain oriented in the same direction. Even if the bullet weren't moving through the air, but just spinning in space, if you tried to push the nose of the bullet to point in a different direction, it will resist you. A spinning top works the same way, that while it's spinning it will resist the force of gravity and remain perfectly upright. Until it starts to slow down.
When a bullet is spinning fast enough to resist the atmospheric conditions it travels through, and it remains oriented with its arc of flight, it is stabilized. When the atmospheric conditions are enough to overcome this resistance and change the orientation of the bullet during flight, it is under stabilized. If the bullet is spinning so fast that the nose of the bullet doesn't follow the arc of its flight as it drops, but rather stays oriented with the barrel of the rifle, it is considered over stabilized.
The bullet resists a change in orientation by using its rotational energy to oppose the external forces being applied - and the spin rate slows down. The greater the external forces applied to the bullet, the more rotational energy is used to oppose those forces, and it slows down even more. A bullet that is minimally stabilized in calm weather may become under stabilized in windy/choppy conditions partially through it's flight path, as it may use up enough of its rotational energy to become effected by external forces as it slows. The farther the target distance, the longer the bullet is in the air, and the more external forces are applied to the bullet. A bullet that is fully stabilized to a 100 yard target may become under stabilized in the same atmospheric conditions out to a 1000 yard target.
Most people think that the biggest factor in choosing a twist rate for a particular bullet at a particular velocity is bullet weight, which isn't correct. Its actually bullet length, but it just so happens that heavier bullets also tend to be longer. The reason is leverage - the longer the bullet the easier it is for the same external force at the tip of the bullet to push it off it's orientation, and the more rotational energy is required to resist that force. Heavier bullets do have the advantage that they have slightly more energy than lighter bullets that are spinning at the same RPM.
There are a number of different formulas and calculators for determining the ideal spin rate for a given bullet. Here is a decent calculator and a brief history of some of the equations used to predict stability:
http://kwk.us/twist.htmlNote that bullet RPM = muzzle velocity X 720/twist rate