In a spectral graph convolution, we perform an Eigen decomposition of the Laplacian Matrix of the graph. This Eigen decomposition helps us in understanding the underlying structure of the graph with which we can identify clusters/sub-groups of this graph. This is done in the Fourier space.
An analogy is PCA where we understand the spread of the data by performing an Eigen Decomposition of the feature matrix. The only difference between these two methods is with respect to the Eigen values. Smaller Eigen values explain the structure of the data better in Spectral Convolution whereas it's the opposite in PCA.
ChebNet, GCN are some commonly used Deep learning architectures that use Spectral Convolution
Spatial Convolution works on local neighbourhood of nodes and understands the properties of a node based on its k local neighbours. Unlike Spectral Convolution which takes a lot of time to compute, Spatial Convolutions are simple and have produced state of the art results on graph classification tasks. GraphSage is a good example for Spatial Convolution.
Additional References: https://towardsdatascience.com/tutorial-on-graph-neural-networks-for-computer-vision-and-beyond-part-2-be6d71d70f49