(I'm a US native speaker). "In" always means inside some boundary. "At" refers vaguely to a place. "At" includes both inside and outside, but because we would usually say "in" if we are inside, "at" usually means outside, but nearby. You need to use "in" if you are emphasizing the "inside-ness." 

Let's say I am driving, I am approaching Logan Airport, and I am talking on my cell phone.

"Where are you?"

"I'm at the airport, I'm looking for Central Parking."

"Where are you now?"

"I'm at the airport, I've just parked and I'm walking toward Terminal B. Where are you?"

"I'm in Terminal B at the Hertz car rental counter. Where are you?"

"I just came through the doors. I'm in the airport now. I'm in the terminal building. I'm in the Arrivals area. Where are--oh, yes, I see you over there, at the Hertz counter." ["At" because there isn't any "inside"].

@Dan Smith has explained the difference between at the airport and in the airport very clearly - while at refers to a general location or point in journey, in refers to being physically in the interior of a building.

Nobody says at airport. This phrase does not exist. @Steven, please don't tell your students that this is British English! It isn't!

'At airport' sounds like pidgin English, or possibly 'telegraphese': for example, you might send a brief informal message to your friend along the lines of  At airport. Plane delayed! Really angry :(  .  If you are speaking in real English, in proper sentences, however, you have to include the article - for example I'm at the airport.

I generally agree with the other comments. "At airport" is very incorrect.

I'm at the airport = I'm in the general area of the airport. (For example, I could be in the parking lot right outside the airport.)

I'm in the airport = I'm physically inside of the airport.

I hope this helps :) 

This sounds like the difference between British English and American English. 

I know of a sort of similar difference between these two dialects of English:

UK: "He is in hospital."

US: "He is in the hospital."