Normally an airline will decide what the allowed maximum wind speed is, for its operations and publish this in its OM-A (general ops manual).
Then each A/C has its set of operating limitations (amongst others, WIND limitations i.e. Head/Tail/Crosswind limits). These are published in the OM-B (the aircraft operating manual with specifics for that type of A/C).
So generally if the wind is too strong and outside the operating "envelope" of the operation at hand, the flight will be delayed till the conditions are favourable or possibly cancelled.
The pilots are trained to the "limits" and above those limits they would be compromising the legal limits. In certain cases the pilots can accept small limit exceedances, but normally this is not done without a VERY good reason and being fully aware of what you're doing. There are certain limits that should never be exceeded, but which might have to be in an emergency situation.