In "Goodfellas", Henry talks about the good times, "before Appalachian and before Crazy Joe decided to take on a boss and start a war". I assume these are Organized Crime - Real Life events/people. Can someone shed some light on this for me. Thanks.

Hill was referring to the mid 50's when the Mob was very strong. In November, 1957 they held a national sitdown in upstate New York (Apalachin) which was raided by NYS police, and soon became a fiasco for the Mob. The meeting was organized by Vito Genovese who was trying to announce himself as the "Big Boss", and to "introduce" Carlo Gambino as the boss of the Family that would soon bear his name (Gambino, backed by Genovese, had the former boss, Albert Anastasia, killed the month before the meeting).

The line about Joey Gallo taking on a boss was a reference to the Gallo/Profaci Wars in the late 50's/early 60's in which Joey Gallo and his brothers were a disgruntled faction in the Profaci Family who sought more money and power from their Family leader (Joe Profaci). This story was the basis for the Rossatto Brothers/Frankie Pentangeli war in "Godfather - Part II".

I knew that the gallo's were based on the rosato's. But in real life did Meyer Lansky backed up the gallo's, like roth did with the rosato's?

No, Lansky had no involvement with the Gallos. At the time of the Gallo/Profaci war, Lansky was very ill, from ulcers and a heart attack, and the strain of having lost all of his Havana gambling interests after Castro's takeover. His next move was to visit Israel, and to look for further gaming opportunities in the newly independent island-nations of the Caribbean. He had no time or energy for the Gallo-Profaci war.