Lossy compression is when, for example, a picture is taken of a field and that photo is then converted into a Jpeg file so that it can be easily compressed down to be a smaller file and can be transferred and shared online, quickly. The file however, when converted and making itself smaller, lost some colour, though to the naked eye remains looking the same although thousands of colours have been lost in the transfer and cannot be recovered. Converting a file to Jpeg isn't a very professional way of converting a file due to the loss of some information in the file.
Lossless compression is when the photo you took (like a field) can be reconstructed from the compressed file, meaning you didn't loose any colour or information from the picture, it is lossless. Lossless files are used to make posters or large scale pictures containing the original information in the picture and are commonly contained in the ZIP files. PNG and GIF files only use lossless compression.