You may not be aware of it, but you're probably using image compression whenever you take a snap with your mobile phone or upload a picture to a social network. The most popular image formats for casual photography and the Web, including JPEG, GIF and PNG, make use of some level of image compression, sacrificing some quality for smaller file sizes. Uncompressed formats, including BMP and TIFF, are more suitable when every pixel must be reproduced exactly without any loss of quality.
Advantage: smaller file sizes
Use image compression on a picture and in almost every case you'll end up with a smaller file size. This means images take up less space on your computer, tablet, phone or digital camera memory card, and if you're working with hundreds of pictures at once then the difference can really begin to add up. Smaller file sizes also make pictures easier to work with in photo editing software -- the smaller the picture, the less RAM and CPU time required to process it, whether you are trying to make alterations through a program on your computer or an app on your phone.
Disadvantage: reduction in quality
The reduction in file size when image compression is applied comes at a cost, and that's a reduction in the overall quality of the image. When using a low level of compression with a format such as JPEG, the difference can be barely noticeable, but this varies depending on the type of image you're working with (the way that light and colours combine in the picture, for example) and the level of compression you've set. Digital photos can often appear very similar even when compressed, because they don't deal with straight, fixed lines and include blocks of pixels of similar colour close to each other (making the pictures easier to compress more effectively).
Advantage: faster transfers
Smaller file sizes not only mean less space taken up on disk and in memory, but faster transfers too -- whether you're loading a picture from your hard drive or displaying it on a Web page, a compressed file will appear more quickly than an uncompressed one. Compressed pictures are almost always used on the Web, where transfer speed is more important than quality, but it can also make a significant difference offline if you need to back up many folders of pictures to an external hard drive.
Disadvantage: variable standards
You'll often be given a choice of how much to reduce the quality of the image when using a compressed format -- this gives you the flexibility to pick the right quality/size balance for your needs, but it also means you might end up with a collection of pictures all compressed at different levels (particularly if you are putting together a gallery using photos taken from other people too). Professional photographers often use uncompressed picture formats or photos that are only compressed very slightly in order to keep the quality consistent across the board.