You generally want wood that has already fallen rather than green wood. If you take green wood from your tree you'll have to let it cure for some time (months) outdoors before trying to use it, though you might be able to cheat and boil the snot out of the thing instead.
Look for wood from deciduous trees rather than evergreens (which have lots of resins), and avoid fruit/flowering tree wood unless you know that the trees haven't been treated for insects in the past year or two. With that in mind, you can use maple, oak, ash, sycamore, poplar, willow, any fruit/nut tree, and many others.
Avoid wood that is very soft and decaying, as it will simply fall apart in the water. Small spots of decay are alright as long as the wood is sound overall though. Generally speaking that means finding wood that is fallen but not covered in leaf litter.
To prepare wood for my tanks, I try to do three things: sterilize the wood, waterlog it, and reduce the amount of tannins it will leach. Sterilization is probably not terribly important, but it makes me feel better and happens incidentally to waterlogging the wood in any case. To do this I boil small pieces in a large pot for half an hour, then change the water and repeat several times. The water should show large amounts of tannins (by being tea colored) but if the wood is particularly well cured it may have already leached away most of them. Boil it at least until the wood sinks, unless you want to silicone/screw the piece onto a stone or ceramic plate. For large pieces of wood you may be able to boil the ends separately, or you may be able to soak it in a garbage can filled with water until it is waterlogged (this takes a while.) You might also just be stuck with having to attach it to something dense and covering that portion with substrate.
Even after you've boiled most pieces of wood, they will still contain substantial amounts of tannins. You can remove more of these by placing the wood in a large garbage can filled with water and leaving it to soak. Change the water out whenever it gets dark, and try to keep the can in the warmest location possible to speed the process. Depending on the type of wood you've chosen, its thickness, and the amount of curing it has already gone through outdoors, this can take a very very long time. If you don't mind the wood discoloring your tank water (some people prefer it) then you can add the wood to the tank once it has been waterlogged by boiling. Whether soaked or not it will likely continue to yellow your water for some time after placing it in the tank. I've got one particular piece of wood (the root ball of a forsythia bush) that has continued to yellow my tank water for over a year now. The effect does decrease over time, and now I only notice it after water changes, but it's still happening.
Before putting the wood in the tank I spray it with a garden hose thoroughly to dislodge any small bits that have decayed or become loosened by the boiling and soaking so that I don't have a bunch of junk floating around in the water.
If you are breaking a piece of wood off of a larger piece, remember that each of the breaks might be visible. If you carefully saw off the individual branches with nice clean cuts and are unable to bury or otherwise hide them, the wood can look very artificial. It's also a good idea to take a bit more of the large piece than you think you'll need, as you can always remove some, but adding bits back on is pretty challenging (though not impossible.)
