From the FAR part 23, the definition of Maximum Operating Altitude:
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(a) The maximum altitude up to which operation is allowed, as limited by flight, structural, powerplant, functional or equipment characteristics, must be established.
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Typically, maximum Operating Altitude is the highest the plane can go for structural/pressurization reasons. For instance, FL370 is our Maximum Operating Altitude in the ERJ because that's the highest we can go will maintaining a 7.8 psi cabin differential pressure while also maintaining an acceptably low cabin altitude (8000 ft), even though the engines and aiframe would have no problem climbing higher if the cabin altitude was not an issue.
Service ceiling, on the other hand, as flyguy said is the maximum altitude at which the airplane can maintain at least a 100 fpm rate of climb with all engines operating.
Absolute altitude is the highest altitude the airplane can reach, period.....in other words you're hanging on at Vy with a 0 fpm rate of climb. Maximum Operating altitude is usally not the same number....MOA is more of an equipment limitation than a performance limitation.