My iPad air will not charge connected to my monitor or my computer. Why?

Short answer:
Computer USB ports don't deliver as much power as the power adapter that comes with the iPad.

Details:

Full size iPads come with either a 10W or 12W power adapter.
The mini comes with a 5W power adapter.

See iPhone, iPad, iPod: Using iPad Power Adapters

http://support.apple.com/kb/HT4327

USB ports on computers deliver 2.5W (USB 2, 500mA x 5v) or 4.5W (USB 3, 900mA x 5v).

Newer Macs with high power ports can deliver 5.5W (1100mA x 5v).

See Apple Computers and Displays: Powering peripherals through USB

http://support.apple.com/kb/HT4049

If you connect your (full size) iPad to a computer USB port, it may charge slowly, but since it's getting less than half the normal charging power, it may say "not charging".
It may charge slowly or it may not depending on use (screen brightness, processor load, etc.).
If you really want to charge it from the computer, put the iPad to sleep.

The best charging solution is to use the Power Adapter that came with the iPad.