PrecisionGoal is an option for various numerical operations which specifies how many digits of precision should be sought in the final result.
PrecisionGoal is an option for such functions as NIntegrate and NDSolve.
PrecisionGoal -> Automatic yields a precision goal equal to 10 digits less than the setting for WorkingPrecision.
PrecisionGoal -> Infinity specifies that precision should not be used as the criterion for terminating the numerical procedure. AccuracyGoal is typically used in this case.
Even though you may specify PrecisionGoal->n, the results you get may have much less than n-digit precision.
In most cases, you must set WorkingPrecision to be at least as large as PrecisionGoal.
PrecisionGoal effectively specifies the relative error allowed in a numerical procedure.
With PrecisionGoal->p and AccuracyGoal->a Mathematica attempts to make the numerical error in a result of size be less than .
See The Mathematica Book: Section 3.9.3.
See also: AccuracyGoal, WorkingPrecision.