Raw satellite images are considered high in resolution, especially multispectral images captured by remote sensing satellites. Hence, choosing the suitable compression technique for such images should be carefully considered, especially on-board small satellites, due to the limited resources. This paper presents an overview and classification of the major and state-of-the-art compression techniques utilized in most space missions launched during the last few decades, such as the Discrete Cosine Transform (DCT) and the Discrete Wavelet Transform (DWT)-based compression techniques. The pros and cons of the onboard compression methods are presented, giving their specifications and showing the differences among them to provide unified information about these methods for researchers and satellite imaging payload designers. Hence, some of these techniques are implemented, and comparisons are presented in the current work as examples to simulate an image compression system on board a small satellite using the MATLAB software package. This was achieved by employing three LANDSAT8, band6 satellite images. A wavelet selection was also considered for the DWT-based compression method, which gave the best results among the other methods through acquiring high values of compression ratio (CR) while maintaining the important scientific information of the image when reconstructed at the ground station.