The Laplacian is a generalized second order derivative. Frequently applied as a filter in image processing. 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 is the sum of the second order derivatives along each dimension, equivalent to the trace of the Hessian matrix.
In image processing, the discrete approximation to the Laplacian most frequently used is a 3x3 filtering mask with weights:
0 1 0 1 1 1
1 -4 1 or 1 -8 1
0 1 0 1 1 1
The left one can be seen as the addition of the 1D second order derivative [1 -2 1] and its transpose. The right one additionally adds two diagonal versions.
The filter can be used to construct an edge detector (Marr-Hildreth), to enhance edges (by subtracting it from the original image), etc.
See also laplacianofgaussian.
