There is a wide variety of software that will allow you to play music on your computer. Most new computers have at least one player installed on them, and in general they will do just fine. However, certain file formats are not supported by all players, which is the main reason you might need to have a few different ones. I have tried most of these applications, but not necessarily the most current version. Click on the product name to link to the website. They are all relatively user friendly and all free.
iTunes
Media player, burner and ripper. Available for Mac and PC. Supports multiple media types. Houses the iTunes music store which is a great resource for legally purchasing music. Free.
Windows Media Player
Media player, burner and ripper. Comes as part of Internet Explorer and installed on new PCs. PC and Mac versions available. Supports multiple media types including WAV, MP3 and MIDI. Free.
Winamp
Media player. Highly recommended. PC, Mac and Android versions available. Supports multiple media types including WAV, MP3 and MIDI. Free.
RealPlayer
Media player, burner and ripper. Can convert video files to a variety of formats. PC and Mac versions available. Supports multiple media types including including RealAudio, MP3, WAV, MIDI and Windows Media. Free.