Nonparametric local polynomial regression estimates a function that is continuous (with no gaps). The fact that the function is continuous allows it to have several derivatives, though the number of derivatives will not be known until the function is estimated, as it relies on the degree of the polynomial function. The benefit of the nonparametric method is that it does not require the researcher to know the form of the function before performing the regression.
The bandwidth matrix method allows the researcher to select a matrix before performing the regression. This matrix should be square (that is, with the number of rows equaling the number of columns) and have a row/column length equal to the number of independent variables. The point of the bandwidth matrix is to give the researcher flexibility in determining the amount of local smoothing that the function will employ. Local smoothing is much what it sounds like: it determines the degree of the "smoothness" of the function at a local scale, making it either more simplistic or complex.
Common in other forms of regression, weighted least-squares methods can be applied to local polynomial regression. The method of weighted least-squares local polynomial regression puts additional constraints on the regression function. This method offers two advantages. First, it lessens the bias of the regression function, giving more applicability. Second, it lowers the variance, increasing the accuracy of the function.
As an alternative to the standard bandwidth matrix method, the automatic bandwidth selection method chooses the bandwidth matrix for the researcher. This method employs an algorithm that attempts to lessen the variance in the data by applying assumptions of structure to the regression function. However, for this method, there are the drawbacks of overestimating the structure in the data and producing an unsmooth regression function.