In most cases, this defect is caused by the chick positioning herself incorrectly for hatching. Normally, one wing will shelter the head inside the shell.
But if a chick doesn't have her wing positioned over her head, the skull can malform, and it will reveal itself within the first few weeks as a cross beak or scissor beak as she begins to grow.
What causes this problem isn't fully understood. Think of it as sort of the chicken equivalent of a breech birth: the chick has just not positioned herself correctly for developing and hatching, and it causes problems. It's theorized that this can occur where the fertile eggs have been exposed to extreme temperatures before incubation, or where the temperature during incubation has varied, either of which means the chick develops off schedule. It can also occur where breeding hens have nutritional deficiencies that affect the contents of their eggs (and thus the nutrition that the chicks use to develop). Sometimes this could be a genetic problem in processing or utilizing certain nutrients, and sometimes it could be caused by a poor diet, or exposure to pesticides or other substances that interfere with normal function. Infestations by internal or external parasites can also cause nutritional deficiencies. In short, a cross beak is just one of the more common "hatch defects" that occur from time to time.
The good news is that in most cases, your chicken can lead a regular, chicken-y life. You will simply have to make sure the feeder and waterer are deep enough to accommodate the way your cross beaked chicken must eat and drink, since it can be more difficult for her. A crossbeak or scissorbeak often needs a fairly deep feed trough so she can scoop out food. Sometimes mixing a little yogurt into her feed will help her ability to pick up the food. Other high protein food like--believe it or not--eggs that have been hard boiled and cut finely can help, too. She will not be able to pick up food with the tip of her beak like most chickens can. She will need to scoop it.
If your bird's case is particularly severe or if she seems to be having trouble eating, a vet may be able to show you how her beak can be trimmed to help her, if that is necessary--although it isn't usually. This is different from the "trimming" done in factory farm operations where practically the entire beak is removed--this simply addresses the tip (like a fingernail, but not to the quick) so the top and bottom beak meet up better.