Short answer
When large mechanical powers are required, AC motors are used, it's not particular to aircraft. The limiting factor of DC motors is the commutator ring. When AC motors are used, AC sources are required. Note this is 3-phase AC, using three phases (over two or over DC) helps simplifying motors: 3-phase motors have a predetermined direction of rotation; they can start without assistance.
Motors and generators are similar devices used in reverse, in one case a current creates a magnetic flux, in the other case a magnetic flux creates a current. The pros and cons of generators and motors are mostly the same, in particular DC generators also need a commutator ring; They have the same power limitation. So the most effective combination is AC generator and AC motor.
AC generators are usually driven by the engines, which rotation rate changes with thrust demand. For a long time the problem has been to stabilize the frequency of the generator using mechanical systems. However today the frequency is no more a problem, we have efficient inverters to electronically stabilize it, so AC generators are also used for small power needs.
I'm detailing the DC motor limitation below, but for your research you may be interested also in these questions:
AC generators are called alternators, DC generators are called dynamos. Dynamos have been replaced by alternators on cars a long time ago. As commented, dynamos don't deliver much current at low speed: We needed to run the engine at a high rpm to actually load the battery on a car with a dynamo. Alternators can load a battery with the engine at idle speed.
Details follow.
Need to invert pole polarity in motors
Motors are based on how two magnetic poles interact: Poles of the same polarity are pushed away, poles of different polarities are attracted.
In a DC motor:
There is a stator, it can be made of permanent magnets or windings. For simplification let's use one permanent magnet with two poles N and S (or two magnets oriented such as the inner poles are respectively N and S)
There is a rotor made of solenoids (windings) within the stator. Let's use a single solenoid. A solenoid fed with DC current behaves like a permanent magnet, creating two magnetic poles N and S. The poles are reverted, becoming S and N, if the direction of the current is reverted.
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Magnetic attraction is used to rotate the rotor solenoid until its poles are aligned with stator poles of opposite polarities. When the poles are aligned they keep their position due to attraction, the rotor stops rotating.
To unlock the rotor, the polarity of either the rotor or the stator is inverted, this changes the attraction into a repulsive force moving the rotor poles towards the next stator poles, where they are locked again. The polarity is inverted again and the cycle repeats.
Commutator ring
Inversion is done by a ring and two brushes connected to the DC source. The ring rotates with the rotor, it has segments connected to the rotor pole solenoids. The brushes deliver the current to the segments.
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As the ring rotates, the segments are in contact with each brush alternatively, the solenoid polarity is constantly inverted.
As the power of the motor increases, the current in the ring segments also increases. But the contact surface between the brushes and the ring segments cannot be increased past a limit, the ring temperature increases with power due to impedance. Wear is stronger, maintenance is required more often.
In addition:
The number of poles is usually large in order to smooth the rotation, meaning the number of segments required is large, and their size smaller, further limiting the maximum current.
Sparks are created between brushes and segments. The effects of the sparks increase with current.
Use AC to remove the commutator ring
The other solution is to change the current direction by directly using AC (AC motors have also other advantages). The inverter is not required anymore.
Advantage of AC generators
Larger aircraft uses AC generators which are more simple and more robust. The frequency is 400 Hz instead of 50 Hz for different reasons, it is mostly historical, and linked to the need to have constant speed drives. With the development of efficient current inverters, the need for constant speed has been removed. One advantage is a higher frequency reduces the size of the voltage transformers since the maximum flux the core can transfer is proportional to frequency.
The 400 Hz frequency was selected by US military long ago as a tradeoff between feasibility of constant speed drives given the engine speed ranges at this time and the corresponding weight reduction (generators, transformers, motors, actuators and wiring, as a higher frequency also requires wires with a larger cross-section due to the skin effect).
Most recent aircraft actually generate AC with a variable frequency, depending on the engine speed. This current can be efficiently rectified, and converted back to AC with the desired frequency, e.g. 400 Hz.
Both AC and DC are present on a large aircraft, but the generators are AC (except the batteries of course). Smaller aircraft which don't require large electrical sources may still have DC generators with a more limited power.