A keel is your underwater wing, just like your sail is the in-air wing. Remember, the sails generate lift around the leeward side of the airfoil, the angle of the boat as it moves through the water causes the keels to generate lift on the windward side of the waterfoil. The vector sum of these forces is what pushes you forward. As I understand it, the advantages and disadvantages are as follows.
A full keel is flat and has a lot of wetted surface to cause friction. They are generally slower and don't generate lift at all. Instead of producing lift they simply provide resistance to sliding directly downwind, which the wind would otherwise blow you. Full keels typically don't go as well to weather as other designs. On the other hand, the draft is shallow and damage is light to non-existent when the boat is run aground. Full keels are very traditional and tend to be very strong too. That's why they are used most often on blue water cruisers where speed is less important than standing up to a harsh environment. They can make maneuvering difficult, especially if the prop is in an aperture.
Fin keels are shaped much more like an airplane wing and do provide lift. The most efficient kind is the high aspect keel. They are narrow and deep and are efficient for the same reason a sailplane's wing is. Because they generate lift, they don't need to be so big -- and so they have less wetter surface and less friction (read fastest). They also put the lead ballast down the farthest, which moves the Vertical Center of Gravity (VCG) down the lowest. That gives you the best righting arm moment to keep the boat more vertical for a given weight. The high-aspect keel also goes to weather the best, but it is the most easily damaged upon grounding, and since it is the deepest draft it is the most likely type to touch ground. Needless to say, you can't get into shallow areas with a deep, fixed keel.
Low aspect fin keels are a compromise. This kind of keel is the shoal draft keel and is the best kind to have where I sail. They are very popular here in the Chesapeake Bay and along the Bahamas, simply because the water is so shallow. Low aspect fin keels don't generate as much lift since the wing chord is stretched longer. And because they don't generate as much lift, they don't go to weather as well and they also have to be bigger, so there is more wetted surface to cause friction. VCG is not as low so they require more ballast. therefore making the boat heavier.
There are a lot of variations on keels -- the full keel with cutaway forefoot. The shoal keel with bulb. The shoal keel with wings. The bulb at the end of the high aspect keel. The lifting high aspect keel. The keel stub with a swinging centerboard, and many more. All have advantages, disadvantages, and some have maintenance actions you need to accomplish every so often. Which kind is best for you depends on the kind of sailing you hope to do and the depth of water you want to get into.
Several before me have spoken about tracking, so I won't cover that.
Hope this helps!
T. P. Donnelly
S/V Tranquility Base
1984 Islander 30 Bahama