Images are made up of many pixels (dots that make up the image). For a basic image, lets say one of 640 x 480 pixels, there are 307'200 pixels in the image.
For each pixel in the image 32 bits of data are required to describe the colour of that pixel.
307'200 x 32 = 9'830'400
So that means 9'830'400 bits are required to describe an image of with size 640 x 480 pixels.
TIFF images are your basic raw image format, so it uses the method described above to store the images. It means that the bigger the image, the more information that is required to describe it and it's pixels. TIFF format is a high quality format that stores the image exactly as it is in the original image.
http://en.wikipedia.org/wiki/TIFF
As you can imagine, as the dimensions of the image get bigger the more information is required to describe that image.
In the mid 80's, a commitee was formed to think of ways to reduce the amount of information needed to describe an image.
The commitee was called the Joint Photographic Experts Group (JPEG). They came up with a new image format named JPEG that could store the same image as a TIFF but using less data.
http://en.wikipedia.org/wiki/JPEG
They developed a mathematical formula called a Discrete Cosine Transform (DCT) that could analyse the image and that could decide which parts of the image were important and which parts the viewer would not be able to notice if they were changed slightly.
When saving an image in JPEG format, the computer stores information about the detailed areas but it encodes less detailed areas using a special algorithm. This changes the image quality slightly but reduces the filesize immensly.
The more quality loss, the more reduction in filesize that can be achieved. This is why JPEG is known as a 'lossy' file format.
Around the same time, Compuserve, an Internet Service Provider were experimenting with ways to store images for graphics. They came up with GIF (Graphics Interchange Format) to save there images. The GIF format also uses a mathematical formula to save it's images, but the formula it uses only supports 256 colours, this means that images saved in GIF format can only be made of up of 256 unique colours.
http://en.wikipedia.org/wiki/Graphics_Interchange_Format
The PNG (Portable Network Graphics) is the next generation of the GIF format. It supports up to 16.7 million colours but it is based upon the same mathematical formula as the GIF format to reduce the filesize.
http://en.wikipedia.org/wiki/Portable_Network_Graphics
Summary:
TIFF
Raw images with no processing or compression. What you see is what you get. Very big filesize.
JPEG
Good for photographs, compresses images anywhere between 10:1 to 50:1 depending upon how much quality loss is acceptable to the viewer.
GIF
Good for graphics such as logos, buttons, icons, etc. Only supports 256 colours but can compress images to a smaller size.
PNG
Good for graphics, supports 16.7 millions colours and is the next generation for graphics images. Can be used for photographs but still not as good as JPEG.
I hope I've fully answered your question. Please rate my answer if you find it useful.