Proofing dough, or allowing dough to rise, begins by activating the yeast with warm water and then checking to make sure that the yeast is alive before adding any additional ingredients. As yeast creates bubbles in the dough causing it to expand, you need to make sure that the dough itself has built up enough gluten in order to hold on to the gas bubbles by kneading the dough. The final step to proof dough is letting it sit until doubles in size, in some cases more than once, before baking it.
For most recipes, you should begin to proof dough by activating the yeast with warm water. The amount of water to yeast that you use will depends on the recipe, although the temperature of the water will never change. Yeast requires warm water between 105° F (about 40° C) to 115° F (about 46° C); anything less than this will not activate the yeast, and anything more can kill it, making it impossible to proof dough. The yeast and water combination should bubble after a few minutes, indicating that the yeast is active and ready for the rest of the ingredients.
Once you have added the rest of your ingredients to the activated yeast, you need to ensure that the dough is elastic enough and has built up enough gluten to contain the gas bubbles created by the yeast. Without this gluten, the dough will typically collapse the bubbles, making it impossible for it to rise correctly. This step is accomplished by kneading the dough, which not only builds up the gluten, but also aids in fully activating the yeast and incorporating it into the ingredients so that you can successfully proof dough. The recipe that you use will typically indicate for how long you need to knead the dough, and you can do this by hand or with a mixer and dough hook.
After the yeast is fully activated and the dough has built up enough gluten to rise successfully, you can proof dough. In order for the yeast to create enough bubbles to cause the dough to rise, it needs warmth. Generally, you can leave the dough in the mixing bowl that you started with, cover it, and leave it undisturbed until it doubles in size. Depending on the recipe, you may only need to proof dough once, although most call for letting the dough rise at least twice. In this case, you will wait for the dough to double in size for the first proofing, and then punch the dough down, shape it into the pan in which you plan to bake it, and let it proof again prior to baking.