Fiber, also colloquially known as roughage, is a type of carbohydrate the body cannot digest, in essence.
That is also where the simple part of fibers ends. Dietary fiber refers to quite a diverse array of compounds. Broadly, these are grouped together into categories according to fermentability, viscosity, and solubility. In general, fibers are comprised of soluble fibers, and insoluble fibers. In case you were wondering the soluble and insoluble here refers to whether these compounds dissolve in water or not.
The reason for doing so is that these factors determine just how these compounds are processed by the body, which is the whole reason that you are encouraged to eat them. Consumption (not that one) is also known to reduce the incidence of certain types of diseases and maladies: constipation, heart disease, colon cancer, diabetes, irritable bowel syndrome, diverticulitis, etc.
As far as sources of fiber goes, they’re the usual suspects. Plant foods and legumes will provide you with varying amounts of the stuff. For ideas on foods not typically associated with being high in fiber, try looking at whole-wheat pasta (which can be a bit pricier) or chickpeas (the canned variety usually isn’t too expensive). If you want to stick with veggies for this purpose, broccoli and artichokes, specifically artichoke hearts, are excellent candidates for fiber.