There is a common belief that if you crop your hair (or shave it), it will grow back thicker than before. There are other, similar beliefs, like the idea that if you pluck out one grey hair, two more will grow in its place.
Conversely, there is also the belief (among some people) that shaving your hair will cause it to grow back thinner, and eventually lead to baldness. Similarly, some people believe that wearing tight hats, or brushing the hair too much, will lead to baldness.
In case you were wondering, none of these ideas is true.
The hair we see on our heads is dead protein. All hair growth takes place in the hair follicle, which is below the skin. When we cut or shave our hair, we're cutting or shaving the dead part of it that has grown out past the surface of the skin. This doesn't affect hair growth in any way, neither positively nor negatively. Hair growth can be affected by environmental factors (stress, diet, radiation, possibly others), but these are things that affect the hair's root. Shaving or cutting hair only affects the dead part of the hair, so it won't affect it's growth.
If shaving somehow did increase hair growth, you can be sure that we'd know about it, because it would essentially be a cure for male pattern baldness. Billions of dollars are spent every year by men trying to re-grow their hair. If all they had to do was shave it for a while, you'd be seeing a lot more shiny domes in the short term.
And conversely, if shaving caused hair to come back thinner or disappear, then we wouldn't need to keep shaving our beards. Granted, beard hair is different from scalp hair, but at the least, those of us who shave our heads would expect hair loss from the scalp to be even, and not focused on the top as it actually is.
However, there may be reasons why people think that shaving can affect growth.
Consider this situation: your hair is receding. You shave it all off and then, after some period of time, you re-grow your hair. You will almost certainly have less hair than before you shaved. Although this is attributable to natural hairloss, it may seem to some people that because they shaved, they lost more hair.
Conversely, one might shave one's head, live the bald life for a while, and then re-grow it without suffering any hair loss or gain. Some people, having become unaccustomed to the sight of their hair, might perceive even more hair than they previously perceived. In this scenario, a person might believe that shaving actually increased hair growth.
While these perceptions might arise naturally, they do not reflect reality. Shaving your head will neither increase nor decrease the number of hairs on your head, nor will it affect their thickness or colour. All shaving will do is to make your hair really short for a while.
Does shaving stimulate hair growth / hypertrichosis? - Includes references to a study done in the 1920's showing that there is no link between shaving and hair growth.
Top Ten Hair Myths - Myth #1 is "Cutting your hair makes it stronger or grow faster."
Hair Follicle Structure - Information about how hair "works".