Prior to getting too involved with describing the many different types of files used for images, it may be good to first identify two major types of images. One being vector type images, and the other being "bitmap" type images. These are considerably different from each other.
Bitmap images consists of an arrangement of pixels "mapped" onto a grid. The digital photo above, to the left, consists of 63,960 individual pixels arranged on a grid (246 pixels wide by 260 pixels high). Each pixel is defined with its own color value. The image to the right is of a small detail from the photo (the very small sign to the right midway down the road) greatly enlarged to show the arrangement of pixels. These types of images can be saved as many different files of types, and can be refined using "photo editing" and "paint" programs.
Vector images on the other hand are very different in how they are defined. Rather than being pixels on a mapped grid, these are defined as an arrangements of points (of theoretically no size) connected by lines and curves (of theoretically no widths) on a two dimensional plain (as in x and y coordinates) or in three dimensions (with the added z coordinate). The good thing is that math needed for defining these images is something that is done behind the scenes for you (by the program you may be using). The first example (above left), shows just the points and curves that make up a drawn object, all of which would be invisible until line characteristics (such as width and color) are added, as seen in the second example. The third example shown (on the right) is with a fill color also being assigned (filling in the voids with green).
Another example of vector images can be found with font characters. Each letter of the alphabet is actually a small vector image. The letter "A" is shown, in the center example, with all of the line and fill characteristics removed. All that remains are the individual dots with connecting lines and curves. The example to the right is with colors re-assigned (using red for the lines with fill of green).
In addition to programs that are specific to vector images (often called "illustrator" programs), there are many other programs that will provide some tools for also working with these types of images.
One of the main advantages of vector images over bitmap images is that they are not size dependent. They are scalable. These images can be enlarged to whatever size is needed (without loss of quality). Another advantage is that the colors can be very specific to select areas of the image. This makes it more adaptable to those types of printing where colors are limited, and for types of printing where gradients and shades of color are not an option. For this and other reasons, logo designs, and files used for printing logos, very often need to be in a vector format.
In time, we wish to expand on much of what has been introduced here. For now though, hopefully this will provide a branching off point on the subject of images, with an understanding of two common - but totally different - ways that images are defined.