A brushless system does exactly the same as an ESC / classic motor combo. Even though their function is exactly the same, these two technologies work in completely different manners. So the vocabulary is also somewhat different:
Literally, "no brush", in opposition to their brushed counterparts. To keep it simple, a brushed motor wears due to the permanent contacts between internal elements (brushes, commutator). In consequence, some motors are rebuildable to allow maintenance service (the SuperStock motors for example) whereas others are said closed endbells (Mabuchi 540, Sport Tuned): you can dispose of these last ones once the internal wear degrades their performance. On the opposite, brushless motors do not wear (so to say) because there is no contact between internal elements: this makes them be considered as "maintenance free" motors.
Brushless system (or "combo")
It means motor + controller. With the brushless technology, especially in "sensored" mode, motor and controller work more closely together, thus a greater preference for "complete" systems specifically built to work together.
Warning: a brushless motor has 3 wires to be connected to the controller. Connecting only 2 cables will immediately toast the motor. Controllers accepting both brushed and brushless motors always have 3 wires too, but only 2 of them are to be connected when using a brushed motor.
Tamiya brushless systems:
I intentionally forgot the Transpeed system (TRF), as well as the first Tamiya sensorless combos (see one on my TA-04 TRF). Both models shown above are sensored, that's what the final S in references means. The second controller can handle less powerful brushless motors, but it is compatible with classic brushed motors (up to the Sport Tuned).
OK, now let's get back to specific aspects of brushless systems:
Sensored / Sensorless
This is an additional wire (or captor) linking the motor and the controller (the left black wire on the second photo). It allows the controller to continuously know exactly where the internal components of the motor are located into the motor cage so the control is more precise. The most noticeable effect is to greatly smooth the motor operations so power is applied much more progressively than on sensorless systems, especially at low speed.
Good to know: a sensored motor must be controlled by a sensored controller. A sensorless motor can be controlled by a sensored or sensorless controller.
Good to know too: by convention, all brushless manufacturers rate their sensorless motors in plain Ts (like brushed motors: 9T, 13T etc). For sensored motors, .5T ratings are used (10.5T, 17.5T etc). To my knowledge, only Tamiya broke this convention with the 54275 TBLM-01S 16T motor.
Sensored / Sensorless comparison
Previously, we talked about Ts applied to brushed motors. Brushless motors also use Ts in quite a similar way: the less T, the higher rpm. But that's where it ends. Besides, there is a performance difference between sensored and sensorless motors.
Here is a document directly comparing Tamiya sensored and sensorless motors. This can be of some help when comparing motors from different manufacturers, not forgetting two manufacturers may announce similar specifications and the motors may offer very different levels of performance.
Brushless motors specifications:
Beyond the T rating, brushless motor performance is most commonly evaluated using Kv. This value rates how many revolutions per minute the motor is capable of when powered with 1 volt. The more Kv, the more powerful motor... or not. In fact, the Kv value rates the rpm (thus the "power"), but it needs to be multiplied by the voltage. So, a 2200Kv motor can rev at 15.840 rpm under 7.2v (a bit faster than a Mabuchi 540), at 16.280 rpm under 7.4v (2S LiPo) or even at 24.420 rpm under 11.1v (3S LiPo). Of course, assuming the motor (and controller) can handle these voltages (most are limited to 2S, either 7.2 or 7.4v depending on the battery type).
The Kv value is important, but the most important is the voltage multiplier. In fact, this is how brushless motors do compete with gas powered units (not to say GP engines are now outperformed), including in heavy categories (1/8th and 1/5th scale).