Machine Learning algorithms can be broken down into two categories, supervised and unsupervised. Supervised algorithms are for problems for which you have a specific quantity you would like to predict (whether or not a phone customer will renew her contract, for example). Unsupervised algorithms are used mainly for exploration and visualization of data, and they are not directly predictive (though they are often used as a preprocessing step for a predictive model).
For supervised classification and regression, some of the most widely used families of algorithms are decision trees and ensembles thereof, support vector machines, neural networks, probabilistic models such as Bayesian networks, and nearest neighbor algorithms.
Unsupervised algorithms comprise a wide variety of different types of algorithms. Some of the common problem settings are association rule discovery, cluster analysis, outlier detection, dimensionality reduction, and various types of reconstructive modeling (such as self-organizing maps and deep belief networks).