A raster image is made of up pixels, each a different colour, arranged to display an image. Examples are GIF or JPG files. Raster images are commonly used for web digital documents and small scale printing. Well, ok. Photos are raster and can be blown up really big for bill boards but those files are really big and very high resolution to make up for the size. A vector image is made up of paths; each with a mathematical formula (vector) that tells the path how it is shaped. Examples are EPS or AI files. Vector files can be scaled to any size with perfect clarity. Besides logos and illustration, vector images also come in handy for stuff like laser cutting, embroidery and packaging die cuts.
What's involved in generating vector images?
That depends on how complicated the design is. If it's just text then it goes pretty quick, unless it's a weird unidentifiable decorative font and the source files for those fonts are not available. Then, just like an image the individual characters have to be traced by hand.
Simple areas of flat colour can be converted to shape fairly quickly by selecting by colour in photoshop, saving the selection as a path and then exporting that path to illustrator. However that path generation tends to be a little loosy-goosy so a bit of anchor adjustment is usually required.
Complicated images, well they just take time. Every shape, every colour, every stroke, every gradient, every bevel, every... every element is traced, colour matched and manipulated "by hand". There is no magic button that you can press to instantly convert a pixelated image into sharp, scalable vector image.
How long does it take?
Graphic design services are billed hourly (to the nearest quarter hour) and it takes as long as it takes. I can ballpark an estimate, I'm pretty good about guess how long a project will take, but don't quote me on that. Below are a few examples of vector projects I've worked on: the fertilizer logo took about an hour, the wrestling girl illustration took about 8 hours.