Image compression consists of reducing the memory size of an image while modifying it as little as possible to maintain a satisfactory quality. What is “satisfactory” depends on the application: the goal is not the same for a consumer photograph and a medical image.
There are two types of image compression. Lossless compression (e.g. BMP, GIF, PNG...) does not change the image but the compression gain is limited. On the contrary, lossy compression (e.g. JPEG) is able to reach a high compression rate but may lose information; this is why lossy compression should not be used for images with few details, like cartoons or logos.