I've been asked many times to explain the difference between these two file types. Artwork integrity really does depend on the file quality, and determining the final application is the best starting point.
Raster files are made up of pixels. Pixels are little cubes of colour linked together to create an image. The number of cubes determines what we refer to as resolution. Once the resolution is set in an image it remains the fixed detail and sharpness. When scaled, these files commonly suffer resolution issues such as
staircasing (jagged edges, refer to this link for illustration - http://joelsbackyardposts.blogspot.ca/2010/11/new-years-resolution.html). There are some programs available to reduce
this however they also have limits. When pushed beyond those limits they will create unsightly artifacts. The colour mode of these images is usually confined to either RGB (Red - Green - Blue) or cmyk (cyan - magenta - yellow - black). Some raster images can have special colours added in Photoshop using a spot colour channel, however this is an expert use of this file format and not accessible to most unless you have a program capable of executing this process. Common file formats for raster images are jpeg (jpg) tiff (tif) Photoshop (psd) Encapsulated Postscript (eps) and Portable Document File (pdf)
Vector files are actual digital line drawings. Because the edges of an image are determined by vectors it is completely scalable to larger sizes and will remain sharp throughout that process. It is also useful to other digital platforms that can read vector files. (embroidery, CAD, CNC) These platforms, unlike printing have no other file formats that will effectively work to reproduce an image.
Vector files often include spot colours as well. Most companies have their logo available in vector file format. This ensures quality control and integrity. A vector file can be easily rasterized to pixels, however to convert a raster file to vector is not as simple. When someone tells you that your logo needs to be digitized, this is what they are talking about. Common file formats include Encapsulated Postscript (eps), Portable Document File (pdf), Adobe Illustrator (ai). All of these file formats can also be raster images as well - they are not guaranteed to be vector. Opening the file in a vector program will help determine usefulness.
This illustration explains visually what a designer sees in preview mode (left, fixed resolution, raster is top image) and then artwork mode (right, scalable, vector is bottom image).