Audacity works with project files; those cannot be opened by other software. In order for others to be able to play your sound clip, you need to export it first.
Preferred format for exporting sound clips will be mp3; mp3 reduces file size considerably allowing for easy transfer over a network, it can be played on any computer. The quality/file size is determined by the bit rate: the lower the bit rate, the smaller the file size but also lower quality. Before exporting, adjust the bit rate settings accordingly (choose Edit > Preferences... File formats tab):
128kbps is well suitable for music
32kbps provides reasonable quality for voice recordings
Once you are finished editing your sound clip, go to File > Export MP3 and select a file name and location for the exported file.
Use this Bit Rate tab to change the compression your mp3 file. A lower number produces a smaller file size. Higher data rates will have better quality audio with greatly increased file size. For lectures, try starting with 32kbps Bit Rate
Audacity Documentation Website.