Flight characteristics - Climb and Ceiling

1."climb rate"
This defines the maximal vertical speed of an aircraft when climbing, while *retaining airspeed*. So no drop in airspeed is permitted for max. climb rate. Climb rate is usually defined in feet/minute

2. "initial climb"
Same as for climb rate, but only for the situation directly after take-off. This figure is important when trees or other high objects are in the direct vicinity of the runway. Aircraft should have a vertical clearance of 15m (45 feet) to any nearby object. So if an aircraft has a poor initial climb, the area adjacent to the runway should be clear of tall objects.

3. "Ceiling"
Ceiling = maximum height (usually measured in feet).
There are two ceilings actually. The operational ceiling (which is the one you probably refer to) and the aerodynamic ceiling.
The first defines how high an aircraft can fly 'normally', thus no drop in airspeed, and reasonable figures like stall speed.
This is where we get to the aerodynamic ceiling. At a certain altitude the aircraft is limited in the 'allowable' speeds. The margin between stall speed and maximum attainable airspeed narrows down to 0 at the so-called 'death man's corner'. A Lockheed U-2 spyplane has an airspeed margin of 30 kts at it's operational height, which is a very narrow margin!

4. "Service Ceiling"
The ceiling at which an aircraft can be flown operationally, which is economically sound or prescribed by the aircraft's mission. As stated, this service ceiling is ridiculously high for the U-2, with little margin for mistakes.

In American usage, "service ceiling" is the altitude at which an aircraft's rate of climb falls to 100 feet per minute. I just skimmed an English book in which they claimed that "service ceiling" was the altitude at which the rate of climb fell to 500 feet per minute. Considering that most WW2 aircraft, other than fighters, have INITIAL rates of climb around 500-700 feet per minute, this would seem a rather harsh standard. Maximum ceiling is worthless for comparisons.

Many German rates of climb are given in meters per second, which I believe is what is displayed on rate of climb indicator gauges (at least modern ones). Comparing an aircraft rated at "22 m/sec" against an aircraft rated at "5.6 minutes to climb to 20,000 feet" is a bit unfair because the second aircraft might have a very impressive rate of climb for the first 1,000 feet.