If there is one thing I'm proficient at its pallet teardown. You have lots of options. It does all depend on the style of pallet you are working with and it some cases its wood composition (hard, soft, pine, oak.....).

For argument sake lets say you are not using the close boarded types.
Just a hammer
Some pallets rely on gravity to hold them together. Putting the pallet up on its side and hitting boards from the bottom can also work. This is truer for staple assembled boards but it can work. Also helps if there is a block of wood to absorb the direct impact and distribute the force more evenly.
Pallet Wrecking Bar.
In some of the more open pallet styles a great way to save as much wood as possible is to use a pallet wrecker tool. You can make your own and there are several now that are commercially available. They all work on the same principal. I had a welder make one for me with some spare angle and square iron I had lying around.

In your searched you can find plenty of variations of this tool. This works great at putting even pressure as close to the nails as possible to pry the board away from the pallet. Super Important make sure if you use something like this that you pry the board from the outside then the middle else you risk a board flying towards your face (seriously. I have smacked my face a couple of times thinking it wouldn't happen). Also, its hard to describe but you need to be sure you place your body on top of the pallet to counter the force you will be putting on it while you use a bar. You want the pallet, minus the board you are removing, to be immobile as much as possible.
Using Weighted Blocks
Before I made my own tool I used the technique from this video. It involves raising the pallet off the ground and resting a block of sorts under the board you are trying to remove. Then, dropping a large heavy object beside that board to push the pallet down thereby raising the board that was propped. Note: Your need to be very careful with your back doing this as there is a lot of heavy lifting. LIFT WITH YOUR KNEES!
Referencing the picture below you can see the brick underneath the board to be removed. The large block of concrete will be dropped on the smaller brick on top of the pallet (beside the board to be removed). That brick prevents the concrete block from damaging the pallet.

Tools to have on hand regardless
Mostly because you will have nails left over somewhere
- Long pry bar: Removing nails from runners and stringers is much easier if you have more leverage. I use a 5 foot pry bar and they pop right out.
- Vicegrips: For when you have stubborn nails or if you pop the heads off when trying to remove them
- Crescent Impact Nail Puller: Another approach for stubborn nails. Very likely with hardwood pallets.
- Saw Horses: Might seem obvious but if you manage to get boards of clean nails can come with them. Using a hammer and some sawhorses is nicer to your back.
Cop out.
Depending on the quality of the boards and how "nail happy" the assembler was lots of people just use a circular saw for the outside (close to the runner or stringer) to get the boards out. Still have to address the nails in the middle using one of the above methods. Your yield will be shorter boards but your risk for splits and cracks goes to null.
This video boasts easy pallet disassembly by just using a reciprocating saw to cuts the nails out entirely. This would leave nails in the stringers and boards so you need be careful of the next tools you use with that wood.
https://www.youtube.com/watch?v=211rNoWrZ-Y