Pulley
A pulley is a grooved wheel with a rope running over it that changes the direction of a force or, in a block and tackle, multiplies it.
Definition
A pulley is a wheel with a rope around it: pulling down on the rope lifts something up, which is easier than lifting straight up. Technically it is a grooved wheel that a rope or cable runs over; a fixed pulley changes only the direction of the force (mechanical advantage $1$), a movable pulley attached to the load halves the force needed (mechanical advantage $2$), and combining pulleys in a block and tackle gives a mechanical advantage equal to the number of rope segments supporting the load. In an ideal system, rope tension $T$ is uniform and equilibrium requires $nT = W$, where $n$ is the number of supporting segments, so $\text{IMA} = n$; real systems lose efficiency to bearing friction and rope stiffness, giving $\text{AMA} < n$. Pulleys also appear in dynamics problems such as the Atwood machine. There, two masses on a string over a massless, frictionless pulley accelerate at $a = (m_1 - m_2)g/(m_1 + m_2)$.
Example
A flagpole uses a pulley at the top: you pull the rope down and the flag goes up. A block and tackle with $4$ supporting rope segments lifts a $400\ \text{N}$ engine with only $100\ \text{N}$ of pull, but to raise the engine $1\ \text{m}$ you must pull $4\ \text{m}$ of rope, so the work is $100 \times 4 = 400\ \text{J}$ either way. An Atwood machine with $m_1 = 3\ \text{kg}$ and $m_2 = 2\ \text{kg}$ accelerates at $a = (3 - 2)(9.8)/(3 + 2) = 1.96\ \text{m/s}^2$, with tension $T = m_2(g + a) = 2 \times 11.76 = 23.5\ \text{N}$. That tension is less than the $29.4\ \text{N}$ weight of the heavier mass and more than the $19.6\ \text{N}$ weight of the lighter one.
Key Insight
One pulley just changes direction; it does not make the load lighter, and adding more pulleys is what lets you lift heavy things with less force. Count the rope segments holding the load, not the number of wheels, because that count is the mechanical advantage of an ideal pulley system. A pulley with real mass has rotational inertia, and part of the system's energy goes into spinning it. Ignoring that is fine for light pulleys, but a heavy flywheel changes the answer noticeably.