A crane is a mechanism that uses a collection of simple machines to both raise and lower objects, and also move them horizontally. They are always equipped with at least a winder; cables, ropes or chains; and sheaves. Both sheaves and the winder are a form of pulley.
Cranes can use a combination of simple machines to gain mechanical advantage and lift objects. The first is the lever, used in the balance-style crane. The crane's beam is balanced at the fulcrum, allowing it to lift heavier objects with a smaller amount of force. The second is a jib-style crane, which uses pulleys to achieve mechanical advantage (but note that all cranes use pulleys--in this case, the pulleys are the main source of mechanical advantage). The third way a crane can lifts objects is by use of a hydraulic cylinder, either directly or in powering a balance or jib.
Horizontal movement in a crane can be achieved in one of two ways. The first is to mount the entire device on a rotating pivot, and simply to swing the load-bearing boom or beam around. This is very common in mobile cranes and also some fixed cranes used in construction. The second method is to roll the load back and forth on tracks along the boom itself. This is very common on the fixed cranes in ports and railroad yards.
A final consideration in how cranes work is stability. A crane is stable when the sum of all movements about the base equal zero. As a practical matter, this means the size of the rated load on a crane must be less than what it would require to tip the crane over. In the United States, a mobile crane can only lift 85 percent of what it would take to tip the crane over. That margin is to account for potential instability arising from what the mobile crane happens to be sitting on.