The Laplacian of Gaussian (LoG) is the generalized second order derivative of the Gaussian function. In image processing, it is applied as a linear filter to obtain a regularized (smoothed) Laplace operator. When using this tag also include the more generic [image-processing] tag if applicable, as well as the language you are coding in for context.
The Laplacian of Gaussian is defined as the Laplace operator applied to a Gaussian function. That is, it is the sum of the second order derivative of the Gaussian along each of the dimensions.
In image processing it is used where a regularized Laplacian is needed: to construct edge sharpening filters, edge detection (Marr-Hildreth), etc.