"I grow plants for many reasons: to please my eye or to please my soul, to challenge the elements or to challenge my patience, for novelty or for nostalgia, but mostly for the joy in seeing them grow." ― David Hobson

There are many ways to acquire material, but the most common way is to start with a trip to the nursery, especially if, like me, you take the greatest pleasure out of developing your bonsai, rather than just having them. Some people prefer to buy a finished bonsai or a pre-bonsai from a bonsai nursery, and that's great; but this post isn't directed to those folks. The principles aren't a lot different, but this post is about how I go about selecting raw stock from a regular nursery.
Picking Your Species. Some people will teach you that the first thing to look for is the species. But I find that advice to be a two-edged sword. It's true that some species are not suitable for bonsai. In general, however, any species that can grow a woody trunk and will survive in a pot has at least some potential for bonsai. If the leaves are small, or you can reduce them, it is better but some trees with fairly large leaves - like grape or sugar maple - have made very good bonsai. If you are a beginner, it is easiest to pick a species that is more commonly used for bonsai, like junipers or Chinese elms, simply because it is easier to get information about the care and maintenance of these trees than, say, a Fukien Tea. But you shouldn't always go to the nursery looking only for a specific kind of tree, because then you'll be tempted to simply buy the best specimen of that kind, whether or not it has good potential for bonsai. Instead, remain flexible and opportunistic. Recently, I went to a nursery looking for some Japanese Maples or Foemina Junipers, and instead left with a cheap boxwood with great nebari, a honeysuckle with very strong trunk movement and taper, and a pyracantha with nice nebari and good lower branches for cutting down to shohin size. The maples had no taper and the junipers in the size I was looking for all had very poor nebari or had other major problems, so I bought none of those.
If you live in a very cold climate, avoid tropical species unless you plan to bring them in during the winter. There are some plants, for example, that are perennials in zones 9-11, but annuals in zones 8 and below. If you are in zone 6 and you plan to keep your trees outside year round, forget those species that are going to die when the temperatures drop into the 30s. If you live in a very hot climate, consider avoiding species that need a couple of heavy frosts in the winter to stay vigorous year round. If you are a novice and don't have a lot of time to spend on your trees, avoid species that require a lot of properly timed attention, like black pine, which needs to be decandled at certain times and needs to have needles thinned at certain times, or wisteria, which needs two well-times prunings if you want them to flower.
Examine the Tree's Health. Some lists of things to look for in a nursery plant will start off with the advice to look for a healthy tree. It is true that you want a healthy tree, but that's not the first thing you look for. Most of the trees you are looking at will be fairly healthy. That's just a given. But you start narrowing your choices down using the criteria below first, then you make sure your tree is going to survive the trip home. Check for strong roots. Check for pest infestations. While you might be able to spray for some infestations, others can be more problematic. You don't want to buy a pine that is infested with beetles. You don't want to put a 15 gallon pot with an ant colony in your back seat. On the other hand, sometimes, you can get really good bargains on trees that look a bit off because some branches got snapped, or the foliage isn't vigorous. Pay for roots, but don't pay for foliage. You can always build the foliage later, and you're going to cut most of the existing foliage off in the next 24 months anyhow. Find a tree with strong roots, a healthy trunk and some movement and build the top of the tree later. Often, you can get that tree home, give it proper soil, proper watering and some good fertilizer and it will start looking like it's on steroids.
Decide on a Budget. Don't start out with very expensive material. If you still don't know what you're doing, and you kill a pre-bonsai black pine that you spent $125 on, you'll feel a lot more discouraged than you will if you take that same money and buy 6 large nana junipers and you kill two of them.
Evaluate the Potential of as Bonsai. Once you've decided roughly what kinds of trees you want, and you get to the nursery and look over some healthy looking specimens, how do you choose the best trees for you in your price range? By evaluating the trunks, the roots and the lower branches.
1. Trunks. I always start with the trunk(s). Look for three things: thickness, taper and movement.
Trunks will thicken over time, especially if you are willing to plant your tree in the ground for a few years rather than in a pot, but if you start with a thick trunk in the lowest 6-12" of the tree, you'll have a huge head start. The heart and soul of most bonsai you will grow from nursery stock will be in the lower 12 inches. Sometimes the lowest 4-6". Everything else, you can grow out and build, but your lower trunk is something you're going to be stuck with 90% of the time.
How is the taper? If you see reverse taper, leave the tree for some other buyer. If you have strong taper, especially near the bottom, you have a better tree. Not every tree in a nursery pot will have spectacular taper, but some will, and those are more desirable. Others can have taper developed later. See how the trunk separates from branches are you move up the tree. Do you see places where you could cut the trunk back and develop a new apex? If not, it'll take much longer to create the desired taper.
How is the trunk movement? If you have very interesting movement in the lower 12 inches, you might have a very good tree. If there are problems with the movement in the lower 12 inches, like a pigeon-breasted trunk at the first bend, this is going to be very hard to correct with pruning and wiring. Move on to the next pot.

You can't see the trunk in this photo, but this was a nice 5 gallon nana procumbens juniper with a good single trunk and a lot of very interesting movement.
2. Roots. Next, examine the roots. There are two things you are looking for - the health of the root ball and the appearance of the surface roots. In the boxwood in my photo above, I liked the trunk's thickness and opportunities to create taper, but I loved the nebari. That's why I bought it. Nebari - the exposed surface roots of the tree at the soil line - takes a long time to develop. It's much easier to spend a couple of hours at a couple of different nurseries examining the nebari than it is to spend five years using advanced techniques to encourage nebari growth. You are looking for roots that will be exposed around the tree at the same level, in all directions, moving away from the trunk and getting thinner as they get farther from the trunk. Roots that loop around the trunk, or come up and out of the soil and then dive back in like a sea serpent are not desirable and sometimes cannot be corrected without endangering the health of the tree. Don't be afraid to dig a little around the base of the trunk. Often, the nebari will be buried underneath an inch or more of soil. Pull that soil away and see how the trunkline will look when you lower the soil level to the point where the nebari flare away from the base of the tree. Sometimes, what looks like a boring tree can actually be very attractive when you remove the excess layers of soil.
Also make sure that the tree has a healthy root ball. If you have a root-bound plant, and you are six months away from a safe repotting time, consider taking a pass. That tree might die before you can get it into good soil. If the roots are soggy or smell bad, consider taking a pass. If the roots are strong on one side, and weak on the other - especially when paired with weak foliage on one side - consider taking a pass.

I loved the based of this Olive, especially after removing some of the surface soil around the base.
3. Branching. Look at the basic shape of the lower part of the tree. If you have a nice trunk and healthy roots with some nebari, start looking at your lower branches. Does the tree have an obvious front? If so, how do the first, second and third branches set up? Ideally, the first branch should be about a third of the way from the roots to the apex (if you appreciate fibonacci sequences in nature, that branch should be about 5/13ths of the way up). If the first strong branch (after any initial pruning you might do) is way too low, or way too high, it might be a poor candidate. If the first branches are thin and weak, can you grow them out and just remove larger branches above? If the branches are totally unsuitable, is it a species that buds back easily? If so, you can build the branch structure later, much more easily than you can thicken a trunk, create movement or develop nebari. If it is a species that does not do much back-budding on old wood (pines and other conifer, especially), the first and second branches are much more important.

This pomegranate has some nebari, and some movement on one of the trunks, but it had a weird belt-like indentation in the lower trunk, and I hated the branching, so I skipped it.
If you've found a good species, in decent health, in your price range, and it has a nicely tapered trunk with some movement, some root flare, a well placed lower branch, a decent branch higher and on the opposite side, and a halfway decent branch above that in the back, you've saved yourself a whole lot of time, work and material. Now get to work building your bonsai.