If you have a color image, multiply the width x height x 3. That is the normal size of the uncompressed image data.
GIF and JPEG are two difference methods for compressing that data. GIF uses the LZW method of compression. In that method the encoder creates a dictionary of previously encountered data sequences. The encoder write codes representing sequences rather than the actual data. This can actual result in an file larger than the actual image data if the encode cannot find such sequences.
These GIF sequences are more likely to occur in drawing where the same colors are used, rather than in photographic images where the color varies subtly through out.
JPEG uses a series of compression steps. These have the drawback that you might not get out exactly what you put in. The first of these is conversion from RGB to YCbCr. There is not a 1-to-1 mapping between these colorspaces so modification can occur there.
Next is subsampling.The reason for going to YCbCr is that you can sample the Cb and Cr components at a lower rate than the Y component and still get good representation of the original image. If you do 1 Y to 4 Cb and 4 Cr you reduce the amount of data to compress by half.
Next is the discrete cosine transform. This is a real number calculation performed on integers. That can produce rounding errors.
Next is quantization. In this step less significant values from the DCT are discarded (less data to compress). It also introduces errors from integer division.