Yeast, as you are thinking of it in this questions, is the rising agent used in baking breads and other baked goods. The process involves mixing the yeast with some sort of sugar; the mixture gives off carbon dioxide bubbles which cause the dough to rise. Moisture of some sort is also required for the fermentation and rising process to take place.
Compressed yeast can also be called cake yeast. It is yeast that still has some moisture in it when it is packaged for sale. It is perishable and must be used by the given deadline for best results.
Dry active yeast is a dry, granular ingredient. Active yeast cells are covered by dried cells and need to be rehydrated before becoming active and able to cause dough to rise.
Quick rise dry yeast, or rapid-rise yeast, has smaller pieces than dry active yeast. This means it is able to dissolve and begin fermenting more quickly. Bread machines make use of quick rise dry yeast due to the short, inflexible time settings involved.