If you want to allow students to record audio and submit it as an assignment - without needing to have and know how to use external recording software - your choices have pretty much been limited to Java-based recorders, which pop up big ugly security warnings when first used, or Flash-based recorders which rely on a streaming server to capture the audio - which typically means an extra server and maybe a hefty license fee for the streaming server software.
Having inherited a solution based on a Flash streaming server which wasn't performing very well (especially under load), I thought there must be a better way in this day and age - and now, there is! I'd like to introduce you to the Online Audio Recording assignment type (currently for Moodle 1.9, 2.x version coming soon):
It requires Flash Player 10.1 or higher, as 10.1 introduced the necessary API to capture microphone data locally. You can record (as seen above) and play back a sound clip, and once you provide a name (as below) and hit the button, it'll transcode it to MP3 and upload it to Moodle via HTTP POST.
When you set up the assignment, you can choose whether to provide a standard HTML upload form for students to submit their own recordings; this simply appears as a basic HTML form and allows MP3, WMA and WAV files to be uploaded.