Compression is used to save space or reduce bandwidth.
Lossy compression means you lose details of the information, but for some types of information that is not a problem. With lossy compression you will not be able to reconstruct the original data, but you'll get something that's 'close enough'. Examples of lossy compression are JPG images, MP3 files.
Lossless compression means you can reconstruct the original data without losing one single bit. You'll need this if the data can't afford to be changed, like a bunch of spreadsheets with financial data. A ZIP file is an example of lossless compression. But in audio you also have FLAC, a lossless compression format for audio geeks.
With lossy compression the amount of compression you can achieve is usually higher than with lossless compression, which is to be expected.