How is the ATC language structured?
I don't know about a formal structure, but ground to air and air to ground communications usually follow the 3-W or the 4-W rule.

For air to ground communication - use the 4-W rule:


Who the controller is (identification of the controlling agency)
Who you are (Aircraft Identification)
Where you are (what is you position and altitude)
What do you want (state your intentions or request)


Example:

Grand Junction Approach, Cessna one seven two Sierra Papa, five miles southwest of the ABC VORTAC at five thousand, five hundred, inbound for Grand Junction to land with ATIS Kilo.

Here the pilot identifies himself as a Cessna with the identification N172SP, that he is five miles southwest of the ABC VORTAC at 5,500 ft MSL, and that he intends to land at Grand Junction airport and has the latest ATIS weather report for Grand Junction.

ARTCC and other air traffic controllers will generally use the 3-Ws to communicate with you


Who you are (the aircraft they are giving their instructions to)
Who they are (the name of the controlling agency)
What they want you to do (instructions for the aircraft in question to follow)


Example:

Cessna one seven two Sierra Papa, Grand Junction Approach, turn right to 030.  Descent and maintain three thousand two hundred.  Expect vectors for runway three zero left.  Squawk six four one one and ident.

Here the controller (Grand Junction approach) has advised Cessna 172SP to turn to a heading of 030 and to descend to 3,200 ft MSL.  He should also expect further heading instructions for insertion into traffic for runway 30L at Grand Junction airport.  The controller also tells the pilot to set his transponder to the code 6411 and ident, so the controller can verify his position.  

As a courtesy, and often requested or implied by controllers to verify the pilot in question heard the instructions and will comply, the pilot will read back the instruction in full to the controller, followed by his/her callsign.

Example:

Right to 030, descend and maintain three thousand two hundred, expect vectors for runway three zero left, squawk six four one one and ident, Cessna one seven two Sierra Papa.

The NATO phonetic alphabet will be used for individual letters to avoid confusion by both parties as many letters can sound similar on a garbled radio transmission. Other oddities include the use of the word 'niner' in place of the number 9; This is to distinguish from the number 5, which is pronouced "fife".

English is also the accepted international language for use in flying and radio communications, but is only required for flights under instrument flight rules (IFR).