What kind of battery has the most capacity for the weight? I want to take my transceiver backpacking, so I care about weight.
Lithium, NiMH, then Lead acid.
Lithium batteries are expensive and potentially hazardous. Among these, Lithium Iron Phosphate, or LiFePO4, is supposedly less dangerous, but still needs a charge/discharge controller. You can find replacements for 12v batteries on eBay using LiFEPO4 and capable of high currents of transmitting but those are still several hundred dollars. If you can settle for around 10 amp hours you can get a lifepo4 cell pack and a controller and charger custom made for around $150-$200. The charger I received charges the pack in about an hour and this is a huge plus and avoids the problem of having time but not being ready to go.
You can string 10 NiMH 10ah D cells in series to make a 12-14v pack. I use an ordinary pvc water pipe as the shell, but a sheriffs deputy once told me it looks kinda like a pipe bomb.... This will cost over $100 and still takes a while to charge. One advantage is that cells can be charged or replaced individually.
Lead acid is comparably cheap, but heavy to lug around. But use agm deep cycle batteries for this, not vehicle starting batteries.
See also the beach portable discussion on my QRZ web page for KI6CQ
Lithium batteries, either Lithium-Ion (LiIon) or Lithium-Polymer (LiPo) batteries currently have the most capacity for the weight. LiPo batteries are very popular in radio controlled cars and RC aircraft - their power-weight ratio is practical for flying things, and they can give out enough current for brushless DC motors, so they should be good for backpacking.
The downside is that they can explode if improperly charged or physically damaged, and when they do, they can easily start a fire. RC folks sometimes charge LiPo batteries out in a balcony, and/or in some sort of container which will let out the blast but contain the fire. A purpose-built LiPo charger which charges each cell individually is a must - they're available in RC stores.
Seach for "lipo explosion" on Youtube for videos. You don't want that happening in your backpack or ham shack, there's a lot of fire involved.