I have a 2013 Mac Air notebook. I'd like to hook it to my TV which has an HDMI port. What kind of a cable to do I need to purchase?

From Apple's technical specifications for your MacBook Air, the following are what video outputs are supported with this device.

Graphics and Video Support

• Intel HD Graphics 5000
• Dual display and video mirroring: Simultaneously supports full native resolution on the built-in display and up to 2560 by 1600 pixels on an external display, both at millions of colors
• Thunderbolt digital video output
  • Native Mini DisplayPort output
  • DVI output using Mini DisplayPort to DVI Adapter (sold separately)
  • VGA output using Mini DisplayPort to VGA Adapter (sold separately)
  • Dual-link DVI output using Mini DisplayPort to Dual-Link DVI Adapter (sold separately)
  • HDMI audio and video output using third-party Mini DisplayPort to HDMI Adapter (sold separately)

Specifically, for connecting your notebook to a HDMI port, you would need a Mini DisplayPort-to-HDMI adapter as noted in the last line.