Does Jing record audio with video? Absolutely. If your microphone isn't working, something's wrong. Maybe your mute button is on? 
Jing looks for the default audio input device. Usually that's the microphone. Check your Control Panel (Windows) or System Preferences (Mac) and make sure that the default audio input device is your microphone. You might need to adjust the volume level. If you need help, check out TechSmith Tech Support.
Why is my audio quality so poor? The likely culprit is your microphone. There is a big difference between using a built-in microphone versus a dedicated microphone. As a general rule, USB microphones are better, and you'll want to keep it away from your computer fan and other white noise if possible. Here are some more tips.
Can I record my system audio? Jing can usually record system audio. Why the italics? Well, especially with Windows Vista there are a lot of "if's" and conditions based on your sound card and more. Keep in mind though--for both Windows and Mac, Jing was not really designed to capture system audio. We focused on voice capture and keeping the audio files small. We should also make clear that you can't switch back and forth between microphone and system audio once you start recording.
While this article was written for Camtasia Studio, the content is extremely applicable to "fixing" your microphone setup and recording system audio.
=Includes video tutorial