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N1 and N2 are the rotational speeds of the engine sections expressed as a percentage of a nominal value.

It is similar to the revolutions per minute (RPM) of a piston engine, expressed in percentage instead of in the actual rate of rotation.

They are different because they are characteristic on two-spool engines (see the figure below). The first spool is the low pressure compressor (LP), that is N1 and the second spool is the high pressure compressor (HP), that is N2. The shafts of the engine are not connected and they operate separately.

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Yes, these need to be roughly proportional for the engine to work, and will move towards an equilibrium if they are not.

The compressor provides the air for the combustion, and injecting more fuel will only generate a little extra power, which however increases the turbine speed, increasing air intake, which will allow more fuel to be burned, again increasing available power and so on.

In practice, if the air/fuel mixture is way off, compressor surges can happen, so in modern systems the engine computer will limit fuel flow.

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    $\begingroup$ It definitely isn't linearly proportional though. Engines usually idle at more than 50% N2 (and 20-30% N1), but that does not mean the core would be producing 50% power or consuming 50% of fuel. $\endgroup$
    – Jan Hudec
    Apr 18, 2016 at 6:25
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Yes. There is a system called the Fuel control system, it's like an intelligent 'tap', regulating precisely the fuel supply to the nozzles for combustion. Its operation is governed by the Engine Control Unit (ECU), and having designed automation systems to test this critical system, I can tell you that one of the variables used to control the fuel consumption is the rpm of the turbine. Especially to avoid a condition known as 'turbine overspeed'. Use this tip to seek more info on the internet (very rare to find even a picture of fuel control systems).

Sorry I can't spill any more info.

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