How does the current and voltage affect the torque and speed of a brushed motor?
Motors are mainly driven by current. The current produces a torque. Then, as the motor speeds up, it produces a back-EMF (EMF = voltage) that causes the current to decrease.

The motor also has a certain resistance. The resistance does not help to spin the motor. The resistance produces heat, which is equal to \$I^2R\$ - or current squared times resistance. The resistance limits the current even when the motor is stopped there is no back-EMF, but usually it limits it way too high.

When you connect a motor to a voltage source (such as a battery) a lot of current flows (because there is no back-EMF) which causes the motor to start turning. The motor speeds up, back-EMF increases and current decreases, until the back-EMF is only a bit lower than the battery voltage. At that point the current is quite low.

While the motor is speeding up it gets hot because of the high current. If used as designed, the current will decrease before the motor gets hot enough to cause any problems. If the voltage is too high or the motor is jammed so it can't spin, that might not happen and the motor can overheat.

We don't know what voltage your motor is designed for, but the most likely problem is that it's designed for less voltage than you connected to it. Or maybe it was already damaged by something else.

If the motor doesn't spin at all when you connect power, it's permanently damaged. Get a new one that you actually know the specs for.