Pixels are a single point in a raster image. It is the smallest part of the picture that can be controlled. Pixels are tiny, individual squares that can be controlled by the user and placed to make a picture e.g the colour and pattern of the squares can make a picture of a computer. Pictures are made up of these pixels which are mostly used in two-dimensional images.
Resolution means how good the quality of the picture will be, how much detail the image will hold, for instance, if the picture has a high resolution then the picture will be smooth and not jagged, whereas if the picture has a low resolution then the picture will be un-focussed and have jagged edges. Resolution is measured in DPi and PPi ( dots per inch and pixels per inch) for example, if you wanted an image to show 8 inches on screen, you would need a resolution of 800 x 800.
A Vector file is a file that is made up with mathematical co-ordinates. The image is made up of points, curves, lines and shapes. As Vectors are made using mathematical co-ordinates the images presented using Vector can be stretched to be as large as you want and not lose quality say, if the image were small, which is good for advertising purposes.
A Bitmap file is an image created by using Pixels i.e. images, photos, snapshots that have high amounts of colour in them. This type of file is not very proffesional and not that good, as when the image is stretched like a Vector, it loses quality and becomes pixelated. This renders it useless unless you intend to leave it the same size and have no need of stretching the image.