NSum—Wolfram Mathematica 9 Documentation

Details and Options Details and Options

AccuracyGoal	Infinity	number of digits of final accuracy sought
EvaluationMonitor	None	expression to evaluate whenever f is evaluated
Method	Automatic	method to use
NSumTerms	15	number of terms to use before extrapolation
PrecisionGoal	Automatic	number of digits of final precision sought
VerifyConvergence	True	whether to explicitly test for convergence
WorkingPrecision	MachinePrecision	the precision used in internal computations

	"AlternatingSigns"	method for summands with alternating signs
	"EulerMaclaurin"	Euler-Maclaurin summation method
	"WynnEpsilon"	Wynn epsilon extrapolation method

With the Euler-Maclaurin method, the options AccuracyGoal and PrecisionGoal can be used to specify the accuracy and precision to try and get in the final answer. NSum stops when the error estimates it gets imply that either the accuracy or precision sought has been reached.
You should realize that with sufficiently pathological summands, the algorithms used by NSum can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NSum.
VerifyConvergence is only used for sums with infinite limits.
N[Sum[...]] calls NSum for sums that cannot be done symbolically.
NSum first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
NSum has attribute HoldAll, and effectively uses Block to localize variables.