The Euclidean distance (also called the L2 distance) has many applications in machine learning, such as in K-Nearest Neighbor, K-Means Clustering, and the Gaussian kernel (which is used, for example, in Radial Basis Function Networks).
You can think of building a Gaussian Mixture Model as a type of clustering algorithm. Using an iterative technique called Expectation Maximization, the process and result is very similar to k-means clustering. The difference is that the clusters are assumed to each have an independent Gaussian distribution, each with their own mean and covariance matrix.
In the Stanford Deep Learning tutorial, whitening is introduced as a powerful pre-processing step for feature learning.
Many machine learning techniques make use of distance calculations as a measure of similarity between two points. For example, in k-means clustering, we assign data points to clusters by calculating and comparing the distances to each of the cluster centers. Similarly, Radial Basis Function (RBF) Networks, such as the RBF SVM, also make use of the distance between the input vector and stored prototypes to perform classification.