2 Easy Approaches:
A) Use an audio recording/editing program on your computer. You will need a microphone that plugs into the computer. When done you will save your recording as a file on your desktop, in your hard drive, and maybe tranfer to a disk.
Here's a wonderful series of videos created for teachers that will show you how to use the most popular, free audio software around - AUDACITY.
TECH EASE - Quick answers to real classroom technology questions.
B) Or you can use a portable, digital audio recorder, generally $59-ish and up. You can get these at places like Radio Shack, Big Box and Dept. Stores, Office MAX/STAPLES type stores, online from sites like NewEgg, Musician's Friend, etc.
Basically, you record with the small hand held device through its built in microphone (some allow external mics to be plugged in) and then connect the device to your computer with a USB cable (the one with the audiot editing software installed) and then edit (if you want) using the software (some of these devices come with software). When finished you upload your audio...
See previous post on Uploading Audio: