Two of my favorite and most popular brands of toothpaste today include Crest Pro-Health and Colgate Total. Both are accepted by the American Dental Association.
Crest Pro-Health includes the active ingredients stannous fluoride and sodium hexametaphosphate. The stannous fluoride protects against gingivitis, plaque, cavities and tooth sensitivity. The sodium hexametaphosphate fights stain and tartar build-up. It is advertised as the most comprehensive protection for your teeth.
The other product, Colgate Total, in addition to having the ADA approval, it is also approved by the Food and Drug Administration (in its effect to fight gingivitis). The proven effects of Colgate Total last for up to 12 hours after you brush your teeth (even after eating and drinking). Along with fluoride, this toothpaste has another active ingredients called triclosan (an antibacterial chemical). Linked with a copolymer, triclosan is able to attach to teeth, providing the 12 hour protection.
Obviously, there is a lot of scientific research going into the development of these products.
Two recent studies include this one for Colgate Total, and this one for Crest Pro-Health.
What it comes down to for each person is a matter of taste. I for one enjoy Colgate Total's taste and texture, whereas I find Crest Pro-Health to have a gritty feeling on my teeth when I brush (although the flavor is very pleasant).
I haven't discussed proper brushing technique yet, however for now just remember to brush your teeth at least twice a day (once after breakfast, and once before bed). Use a fluoride-containing toothpaste, and consider giving one of the two listed above a try!