Exercise has been shown to increase attention span in kids, reduce depression, boost memory in middle-aged and older people, and increase ability on complicated mental tasks. Research has also shown that regular exercise helps protect your brain as it gets older particularly helping memory and intelligence (they often use the word cognitive ability). For example, exercise decreases the rate of diseases like Alzheimer's. The findings have been quite dramatic, but it is important to remember that this research is done under specific and controlled conditions.

That said, it is probably safe to assume, given the wealth of studies, that regular exercise is good for your brain and helps keep you sharp. There are many theories about how this happens; one theory, particularly for the protective effects in older people, is increased blood flow from exercise also gets extra blood to the brain. However, anything involving phenomena like intelligence and memory is very complicated, so there are likely many things that take place.

addendum to this answer
It seems that a robust study published two days ago (June 6th, I know!! Super recent!) found that exercise does not help with depression. The results were very surprising to researchers in the field, and the study has, of course, generated considerable debate, particularly among sufferers of depression who have found exercise very helpful.

Yes, physical health can affect your ability to learn in many ways! In general, being in better physical health means better learning. This goes for both illness and physical fitness. First, being sick affects your ability to learn because it's so distracting. The human brain can only do a limited number of things at once, so if you're using your brain to think about how your body aches and you need another tissue, then it's very difficult to also concentrate on homework. Even when you try not to think about how bad you're feeling, your brain may still be distracted. Researchers have shown that having a cold makes you less alert and slows your ability to react to things. Colds and other respiratory illnesses also make it harder to pay attention to something for very long.

Second, being active greatly increases your ability to learn. You may have heard that the human brain can't grow any new brain cells. This is generally true, so you should take good care of your brain! However, researchers have recently made a fascinating discovery-- you CAN grow new brain cells in one part of your brain. The dentate gyrus of the hippocampus is a part of the brain involved in forming new memories. Scientists have shown that you can grow new cells in this part of your brain by exercising! So next time you're studying for a test and feel like you can't learn anything else, go outside and exercise. You'll help your body and your brain!

Cotman, C. W., & Berchtold, N. C. (2002). Exercise: a behavioral intervention to enhance brain health and plasticity. Trends in Neurosciences, 25(6), 295-301.

Hall, S., &Smith, A. (1996). Investigation of the effects and aftereffects of naturally occurring upper respiratory tract illnesses on mood and performance. Physiology and Behavior, 59 (3), 569-577.

Smith, A., Thomas, M., Kent, J., & Nicholson, K. (1998). Effects of the common cold on mood and performance. Psychoneuroendocrinology, 23(7), 733-739.

The answer is yes! If you are healthy, your body works better and is more alert, which makes learning much easier. When you are sick, your body spends more energy on trying to get better. This leaves less energy for other things like learning. The same can be said for people who are physically fit. Science has shown that people who exercise and eat healthy learn better. The following article explains some of the reasons why exercise helps your brain grow and learn:

Yes, absolutely - remember that your mental attributes are a product of your physical body, particularly your brain and nervous system. If your brain and your nervous system are getting plenty of oxygen and sugar and the right hormones, they will function more effectively, but only if you are in good physical health. This is why saying that 'the brain controls the body' is something of a mislabel.