Lithium primary batteries (both AA and CR123) are outstanding in terms of cold/hot performance, energy density, and shelf life. Probably the best there is of any chemistry. Their chief drawback is price. At $1-$2 each, they are not good for "putzing around" with your sweet new light, and of course they have to be thrown in away when they are used up.
Li-ion rechargeables have nearly the same properties, but can be recharged. Their drawbacks are 1) very limited availability and 2) fire hazards from over discharging and charging. Obviously, these are the cream of the crop since pretty much every portable electronic device uses them (laptops, cell phones, etc). 14500 = AA size (but 3.7 volts instead of 1.5 volts). A 14500 will barbecue an AA device that isn't designed for the higher voltage. Many flashlights are designed for them though, but by no means all (Fenix AA lights for instance are NOT, while Quark, ZebraLight, JetBeam, and NiteCore among others are).
Low self discharge (LSD) AA NiMH rechargeable batteries are a superb compromise between Li-primary and Li-ion. They are cheap, easy to find, and have pretty decent cold/hot properties. They will work in any AA rated device. They also have a moderately long shelf life (1-2 years). They are generally sold/labeled as "Pre-charged" ready to use rechargeable.
Regular AA NiMH rechargeable batteries have better energy density (longer run times) than LSD NiMH cells, but they self discharge rapidly, and will go dead within a few weeks/months. Great for constant use devices, but not good for long term, low use devices in which a battery might sit for many weeks/months.
Hope that clears up some things. For a single AA light, be sure to check the "operating voltage" stats. Anything that says it can handle a maximum of at least 4.2 volts will handle every AA size battery. If you don't intend to invest the cash in a Li-ion rechargeable setup (~$40) then I wouldn't worry about it much, and I'd go with NiMH LSD cells and a cheap charger (Sanyo Eneloop, Duracell, Rayovac Platinum).