In general, any technique that can enable a machine (i.e. usually implemented as a computer program) to "learn" from a dataset and apply that knowledge to new or unseen instances of data, could qualify as a machine learning technique. One important performance measure of any machine learning technique is its ability to learn from specific instances and generalize that learning to new instances. There are several types of machine learning algorithms (supervised and unsupervised learning, reinforcement learning) and techniques/approaches associated with these algorithms. A few examples of machine learning techniques are Neural nets with backpropagation, Bayesian networks, Regression, Support Vector Machines, Self Organizing Maps, k-means clustering, Deep belief networks and Boltzmann machines. An artificial neural network would definitely be a type of machine learning approach.
Machine learning = (Statistics - Assumption Checking) x Computer Science x Content Expertise
ANN is only one of the many techniques available for machine learning, as mentioned by Naresh. Other names often used interchangeably are data mining, statistical learning and predictive modelling.