I'm having a MacBook Air (Late 2010), how can I use AirPlay with AppleTV?

If you are referring to AirPlay Mirroring, it requires a Mid 2011 or newer MacBook Air, so you can't use AirPlay Mirroring.

Instead, there are third-party applications that allow you to mirror the MacBook's display onto the Apple TV without needing AirPlay Mirroring, like AirParrot and Beamer. Make sure the Apple TV and MacBook Air are connected to the same network

I had the same issue. On the Tech Specs page for Mavericks, it clearly stated that late 2010 Air's were supported. However, today the page has been changed;

AirPlay Mirroring

Requires a second-generation Apple TV or later. Supported by the following Mac models:

• iMac (Mid 2011 or newer)
• Mac mini (Mid 2011 or newer)
• MacBook Air (Mid 2011 or newer)
• MacBook Pro (Early 2011 or newer)
• Mac Pro (late 2013)

So you aren't going to get AirPlay working. Your only option is something like AirParrot, as mentioned by mende1.