In general, circuit breakers provide protection from overloads and short circuits. Their primary purpose is to protect the wiring - overloaded wiring will overheat, break down insulation, and eventually start a fire.
They are designed to trip immediately on short circuit, when you have line to line or line to neutral contact. With a short circuit, you have extremely high current.
They are not designed to trip immediately on overload at their nominal protection rating; in other words, your 60 amp breaker won't trip immediately under a 61 amp load. There is a "trip curve" for the breaker, which describes how long it takes to trip on overload - longer / slower for a slight overload, shorter / faster for a more dramatic overload.
When accidental contact causes current to flow through your body, your body's resistance is high enough that the current is usually quite small; it only takes tiny currents to cause injury or death. This won't trip a regular breaker - it isn't a short circuit or an overload. Regular circuit breakers are not designed to trip for safety.
Note that faults that travel through ground are also usually through high enough ground resistance that small currents flow, not enough to trip a breaker. When an equipment grounding system is in place, it will provide a low resistance path for many faults, allowing enough current to flow to quickly trip the breaker and "clear the fault."
GFCI / GFI / RCD circuit breakers have another means of tripping. A GFCI device monitors for the current on each of the two or three wires to see if there's any current going out on one wire that isn't returning on the others, which means there must be current flowing through ground, which could be a person receiving a shock. There are different types of GFCI, but common US class A GFCI trips at only 4-6 milliamps of current, low enough to prevent injury. Other types have higher trip levels.
Note that a GFCI only protects against accidental contact that causes a ground fault; under some circumstances you can still receive a shock from a GFCI. For example, if you are wearing rubber soled shoes on a dry wooden floor and are making no ground contact, and you grabbed a bare neutral with one hand then a bare line conductor with the other, there would likely be no current flowing through the ground, and the GFCI would not trip.