Sometimes this is quick, easy and painless. Other times, it's very time consuming and can get expensive. Going from a line drawing (vector file) to a pixelated image (raster) is usually easy and inexpensive. Going from a raster file to a vector file demands a trace (which is usually not very precise) or a complete redraw. It can cost a lot of money. In some cases, it's impossible.
Some images cannot be interchanged. For example, a photograph cannot be a vector file. It just can't. Photos are made up of lots of little pixels and when viewed all together, make the image. You just can't draw that. And you also can't make a die-cut vinyl out of it, or send that image to a merchandising company to print your logo on a golf ball.
When working with a designer, be sure to talk to them about how the art is being created, and what it can be used for. Discuss this in the planning phase, so that when you get the files, they're exactly what you need. And try to plan for every possible use for your images up front. It's much easier, and affordable, to get them in the right format in the first place than to have to have them redone for different uses down the road!