The bases and rings used to connect a riflescope to a firearm are often the weakest link in a shooting system. Without a properly fitted connection, the very best scope and rifle combination is rendered useless. Often it is impossible to tell if you have an unsecure or ill-fitting connection until you're at the shooting bench or in the hunting field, where a poor mount or installation may show itself as inconsistent at best, and at worst cause an excess expenditure of valuable time and ammunition or a wounded animal. Think of scope mounts as the lug nuts of the shooting world.
Once installed you should never have to trouble over a loose wheel. Choosing rings and bases is not the time to pinch pennies or just make due with "good enough." The money spent will be more than made up for in reliability, ruggedness, and peace of mind. Glossary: These terms are thrown around everywhere quite loosely. Take them with a grain of salt.
Picatinny/1913 rings and bases are similar to Weaver style. The main difference is that they are based on specifications standardized by the U.S. Picatinny Arsenal in 1913. The main physical difference is in the size of the recoil lugs on the bottom of the rings and the corresponding larger size of the recoil slot in the top of the base. This size is .206", much bigger than the .180" in the Weaver style. Therefore, a Picatinny ring will not fit into a Weaver base, but a Weaver ring will fit into a Picatinny base. Some manufactures label their products as Picatinny/Weaver but are actually Weaver style. Some Picatinny rings are tightened with an inch-pound torque wrench to ensure a return to zero when reinstalled. Our military M-4 carbines utilize Picatinny rails.
The Leupold style of bases (also Redfield, Burris, and others) are a standard that non-Weaver style mounting systems are measured by. They can be one or two pieces, and are steel, sleek, and strong. They are reliable and trouble-free. There is almost as much interchangeability between manufacturers with the Leupold style as there is with Weaver style systems. Unlike Weaver style rings, the Leupold system is not easily detachable. The top half of the rings must be separated to remove your scope. The front ring is kind of a press fit, with a protruding, beveled rectangle of metal under the ring, turned tightly into a corresponding dovetail slot in the front base. Normally, the two halves of the rings are loosely assembled, and a scope ring tool or a one inch wooden dowel or a screwdriver handle is inserted between the pieces to gain leverage to turn the ring into the base. This is a press fit and cannot be done by hand.
The rear ring sits flush atop the base and is held there by two opposing screws tightened into it. The screws have a leading edge that fits into corresponding slots in the ring. The base screws are tightened into each other, squeezing the ring between them tightly and immovably. By backing out one screw and tightening the other, the ring moves laterally on the base, effectively acting as a windage adjustment. This is of minimal importance unless your base holes are drilled off-center, or there is some other problem that makes your gun shoot left or right so much that it can't be easily corrected with minor internal scope adjustments.