Raster graphics are a collection of various colorful pixels that, in combination with each other, form an image. They are typically filed under GIF or JPEG titles.
Vector files are a collection of curved or straight linear paths. Instead of being comprised of pixels, they contain data, such as curvature and color fill percentages. They are typically filed under AI or EPS titles.
Due to their lack of pixels, vector images can be scaled to be very large while maintaining their quality. This makes them ideal for commercial use. Raster graphics lose quality at a much quicker rate because their pixels are forced to increase in size with the scale change. This makes them more suited for personal projects.