1. June 15, 1785 – Jean-François Pilâtre de Rozier attempted the first manned crossing of the English Channel in a balloon but tragically crashed.
2. June 25, 1905 – Wilbur and Orville Wright made their first flight of 1905 in Huffman Prairie, Ohio, in their new “Flyer III,” the first practical airplane in history. 
3. June 26, 1909 – The first commercial sale of an airplane in the United States was made as Glenn H. Curtiss sold one of his planes to the Aeronautic Society of New York for $7,500. This action spurred the Wright brothers to begin a patent suit to prevent him from selling airplanes without a license. 
4. June 14, 1919 – Alcock and Brown completed the first nonstop transatlantic flight in a Vickers Vimy aircraft from Newfoundland to Ireland.  Most people think Lindbergh was the first to complete a transatlantic flight in 1927, but Alcock and Brown were first.  Lindbergh was the first person to complete such a flight solo, and first to fly from York to Paris nonstop.
5. June 17, 1928 – Amelia Earhart became the first woman to fly across the Atlantic Ocean as a passenger.
6. June 6, 1944 – A huge airborne armada, nine planes wide and 200 miles long, carries American and British troops across the British Channel for the D-Day invasion of Europe.
7. June 26, 1948 – The Berlin Airlift began, where Western Allies supplied West Berlin by air during the Soviet blockade.
8. June 18, 1983 – Sally Ride became the first American woman to travel to space aboard the Space Shuttle Challenger.
9. June 1, 2011 – Lufthansa introduced the Boeing 747-8 Intercontinental (747-8I), marking a milestone in commercial aviation.

Above is an abbreviated list of a few of the aviation events and milestones that occurred in June.  For a more complete list see this Wikipedia page.