First off, defining some of the types and this is not an exhaustive list but only a start.
Primary batteries.
These would be your disposable use-them-once-then-throw-them-away type of batteries. Duracell, Engergizer, Rayovac etc. They include alkaline, heavy duty and the like. Six years ago I did a test that I'll include here for an example and reference. Mainly D, C, AA, AAA but include other types like the button, watch and other styles of battery.
http://budgetlightforum.com/node/36316
Snipits of the test:
The last couple of years or so I’ve been increasing my supply of batteries, both rechargeable and consumable. Always looking for a bargain (while keeping value). I did the following test for grins and to see personally what would happen. I used two Maglite AA mini-mag with incandescent bulb to test them in. Here’s what I got:
Sunbeam Heavy Duty (Dollar Store non-alkaline) Continuous run time: 2:10 (dim at 1:30) 15 cents per battery normal price ($3 for 20)
CVS Brand Alkaline
Continuous run time: 4:58 (dim at 4:20) 49 cents per battery normal price ($11.79 for 24). BOGO brings it down to 25 cents per battery.
Harbor Freight & Tools Thunderbolt Alkaline
Continuous run time: 5:00 (dim at 4:50) 25 cents per battery normal price ($5.99 for 24). Sale price for this test ($4.29) brings it to 18 cents per battery.
Duracell Alkaline
Continuous run time: 5:20 (dim at 4:50). 42 cents per battery based on Walmart price of $9.97 per 24.
I’ll normally get the CVS alkaline batteries when they’re BOGO. But lately I’ve also been getting the HFT Thunderbolt batteries when they go on sale or I have the 20% coupon (which I always have a plethora). I was at Lowes yesterday and saw their Utilitech batteries. Never used them so I bought a 48 count box (and the box was a nice plastic carrying case) for $13.17. That’s 27 cents per battery so it’s about the same as the CVS brand and cheaper than Duracell. Only the Thunderbolt does a bit better and they came in second of the three alkaline brands I tested.
Rechargeable batteries:Sunbeam Alkaline (25 cents per battery)
Continuous run time 4:54 (dim at 3:24)
Fairly impressed with this battery for the price. Not quite as good as the Thunderbolt or CVS but not that far off.
So far the ranking based on run times stacks up like this:
1. Duracell Alkaline – Continuous run time 5:20 (dim 4:50) (cost for test 42 cents per battery)
2. Harbor Freight & Tools Thunderbolt Alkaline – Continuous run time 5:00 (dim at 4:50) (18 cents per battery)
3. CVS Brand Alkaline – Continuous run time 4:58 (dim at 4:20) (25 cents per battery if BOGO)
4. Sunbeam Alkaline – Continuous run time 4:54 (dim at 3:24) (25 cents per battery)
5. Utilitech Alkaline (Lowes store brand) – Continuous run time 3:51 (dim at 3:30) (27 cents per battery at normal store price.)
These are batteries that you can recharge over and over again. In the long run they are more practical and cost efficient. It pays to get the best over junk.
In my experience, the two best I've used are Eneloop and Engergizer as far as AA and AAA batteries. Eneloop can be found in nice packages at places like Costco that often come with a charger. And of course Amazon and other retailers have them. These are probably the best overall from my research and can be recharged like 2100 times (depending on the battery), have a good amount of MaH (capacity of energy) and are advertised to retain 70% of their power level after 10-years of non-use.
I use Energizer in my Taser at work. They are rated for 100 5-second cycles and the power levels seem to last a LONG time. I normally only charge my taser a couple of times a year. And I routinely (as required by policy) to a 5-second cycle test at the beginning of each of my shifts.
Lithium Ion:
Lithium ion batteries have designations like 14500, 18650, 18350, 26650 etc. Many of these designations correspond to a particular battery type. For example;
10440 = AAA
14500 = AA
18650 = 2xCR123
Warning!
While a 14500 battery is physically the same size as a AA battery, it can't always be put in a device that uses a AA battery. The device, such as a flashlight has to be rated for lithium ion battery usage. The reason is that a AA battery is 1.5 volt while a 14500 battery is 3.7 volts. So if the device is not rated for lithium ion battery usage your going to damage it. And possibly cause injury to the user.
On the other hand, many devices like flashlights are often rated for both. For example, the little Sipik or SK68 flashlight that you can get on Ebay or Amazon by the gazillions is rated for both AA and 14500. As a ballpark, if the Sipik is using a AA battery it will provide around 80 lumens which is quite good for such a small LED flashlight. However, if you put in a 14500 battery you'll likely crank up the lumens to 300 which is substantially brighter.
Second Warning!
While you can skimp on regular batteries i.e get the cheap dollar store D, C, AA and AAA and only sacrifice run time i.e. your flashlight or other device won't run as long...DO NOT SKIMP on lithium ion batteries...EVER. The reason is one of safety. Garbage lithium ion batteries can 'explode' during recharging or if improperly recharged. I don't necessarily mean explode like a grenade, but they can 'pop' violently and discharge gases and liquid that are acidic.
The action starts at around the 3:00 mark.
A LOT of lithium ion batteries are made in China. Not necessarily a bad thing, however, China brands are notorious for two things i.e. mAh and lumens. Often called Chinese-mAh and Chinese-lumens. For example, a GOOD 14500 battery will typically have a capacity of around 750 mAh. A few perhaps as high as 900 maybe. If you look on Ebay you'll see crap like 14500 batteries listed as holding 1800 mAh for a buck each. Sounds great! After all it is more than double the GOOD brands and a LOT cheaper! Problem is that you can't fit that level of capacity in a 14500 and in all likely hood it's safe to take off the last zero for a more accurate capacity of 180 mAh. Worse yet, a good li-ion battery should get around 500 recharges, these will likely be around 5 and are unsafe as well like in the video I posted above. So stay away from blue label 'Surefire' brands and anything that has an exaggerated mAh capacity listed. That's to entice the suckers.
You need to go with a trusted label from a trusted supplier. The one I personally use is Mountain Electronics run by a man named Richard who is a member of Budget Light Forum and has an excellent reputation.
https://www.mtnelectronics.com/
This is the source I get all of my li-ion batteries from and I haven't been disappointed yet.
More to follow and others can detail their thoughts and experience as well.