With this list, we're assuming you already have things like measuring cups, spatulas, and long-handled spoons for mixing. For these kinds of things and other baking essentials, check out our round-up of tools for baking prep.
Large Mixing Bowl - A large, even over-sized, bowl is exactly what you need not only for mixing doughs, but also for holding the dough while it rises. With a large bowl, you won't be worried that the dough might spill over the sides! You can certainly find and buy specially-designed rising buckets for this step, but we prefer to keep things simple and just use a big bowl.
Electronic Scale - An accurate electronic scale is absolutely essential once you get into more advanced bread baking and need to maintain ratios of water, flour, and other ingredients. We use ours almost daily for both bread baking and normal cooking.
Bench Scraper - Whether you're just dividing dough into loaves or portioning out individual dinner rolls, a bench scraper makes the job fast and easy. The blade is sharp enough to cut through dough without fuss, but is dulled so you won't risk gouging your counter top.
Standard Loaf Pans - Buy two of these because almost every recipe you'll come across will make two loaves. The standard size is 9x5 inches, and we really like the ones that are made of heavy steel and have nonstick coating.
Proofing Basket - Ok, we suppose this isn't absolutely essential, but they sure are fun to play with! If you want to try your hand at round country loaves, proofing baskets will help the loaves keep their shape and rise higher than they would otherwise.
Peel - You use this long paddle for shuffling round loaves, pizzas, and other breads in and out of the oven. You can find them in made of wood or metal, but look for ones with a handle at least as long as your forearm. The long handle makes the peel easier to work with because you can use both hands to carry and maneuver heavy loaves.
Baking Stone - This is a large, flat tile that sits right on your oven rack and gets heated along with the oven. You can bake loaves (and anything else!) right on top of it. Baking stones are about as close as we home bakers can get to a real hearth oven.
Note that the one thing that we didn't include in this list is a standing mixer. It's certainly convenient to have a standing mixer, but one is only really necessary for a very few kinds of specialty breads, like ciabatta. Don't let not owning a standing mixer be a deterrent for you! Hand mixing and kneading takes some practice, but it's entirely do-able.
Is there any other equipment you like to have handy when you bake bread?