Q: I bought an Ipad Air with a SD Card Reader. I can't read nor record the AVCHD (.mts) movies. Why ?

The Ipad Air and the SD card import adapter doesn't support direct AVCHD import.  It does however support MP4 import directly from your Sony Camera.   I have a Sony Camera which allows me to select the recording format as either AVCHD or MP4.   When I select MP4 at the highest definition...it works just fine, sees the file and allows me to import it into my Ipad Air.

The difference between AVCHD format and MP4 is described in other replies..but in general

AVCHD - A bit higher resolution 1920x1080 and 60 fields/sec, and includes dolby AC3 in AVCHD format audio.

MP4 - A bit lower resolution at 1440x1080 at 30 frames/second. AAC audio in mp4 format.

If you want to do import to the IPad Air and do editing on the Ipad, then you can do so..but only if you shoot in MP4 format on your Sony camera.

If you shoot in AVCHD, and then take the SD card out and  import it into a laptop MAC PRO or desktop iMac in Iphoto..it will import either format.  I've tried shooting both formats on my camera and they both imported just fine on the larger "computers".  Once in the computer, then you can directly drop the videos (either format) into Imovie.

I'm not sure why Apple chose NOT to support AVCHD in the iPad products...but it could be due to the added processing required for 2 x the frame rate.   And frankly, if you look closely you "might" see the difference on an iPad movie with MP4 compared to a "desktop" movie with AVCHD....but it all depends upon the end viewing device.