Cells breathe? Well, not really - at least not in the same sense that people breathe. Despite this, there is a process in a cell known as cellular respiration. It's also called oxidative metabolism, depending on which particular textbook you may be learning from. Whatever you call it, this is a very important process by which a cell generates energy. Let's take a look at some of the mechanisms of cellular respiration.
The basic concept behind any of the mechanisms is to take basic nutrients and produce energy from those nutrients. Chemical energy is usually stored by a cell in the form of a molecule called ATP. More ATP means the cell has access to more energy for growth and repair.
For human cellular biology, the process of aerobic respiration is by far the most important mechanism of cellular respiration. These processes require oxygen to proceed. One of the more important steps in the production of energy in a cell is glycolysis. This involves the breakdown of glucose into pyruvate. The exact process involved in glycolysis is somewhat complicated, but the end results is the production of some ATP and pyruvate.
The pyruvate created during glycolysis can then be sent to another cellular reaction - the Kreb's cycle (sometimes called the citric acid cycle). This is a biological energy producing mechanism that gives severe headaches to biology and medical students worldwide. It's quite complicated and involves eight different steps. The end result of the Kreb's cycle is the production of even more ATP and other high energy compounds.
These other high energy compounds are then passed along by the cell to the last of the major aerobic mechanisms of cellular respiration - the electron transport chain. This step takes place in the mitochondria of the cells, and yields a relatively HUGE amount of energy compared to the previous two steps. Then end result of the electron transport chain is a large amount of ATP, and water. Yes, plain old fashioned, everyday water. And lots of it.
If oxygen is not present, there are still some ways that a cell can create energy, but they are vastly less efficient compared to the above mechanisms. Fermentation can produce some energy without oxygen. Unfortunately, you are also left with nasty byproducts like lactic acid and ethanol. Of course, sometimes fermentation is used to intentionally make ethanol, as in the case of wine and liquor manufacturing.
Bear in mind that this is but the most basic description of the mechanisms of cellular respiration. A detailed study of these metabolic pathways requires a great deal of study. Entire textbooks have been devoted to the details. Hopefully this gives you a start with some of the basics.