Training (Supervised)

The phase of adjustment of the parameters of a predictive model is called supervised training.

The term is more particularly used in the context of supervised Neural Networks, like, for instance, the Multilayer Perceptron (MLP) as used in regression. The function materialized by the MLP should "match" the data (or rather, the true regression function) as closely as possible. A supervised training algorithm will iteratively modify the numerical values of these parameters (also called "weights" in this context) so as to progressively reduce the prediction errors of the MLP, until no further improvement is possible.

The term "training" comes from the prehistory of Neural networks, and its psychological connotations are somewhat amusing to straight statisticians. Strictly speaking,  adjusting the coefficients of a Logistic Regression or developping a Decision Tree could be construed as "training", although nobody in his right mind would call it that way. 

Training (Unsupervised)

Just as Supervised Training adjusted the parameters of a predictive model, unsupervised training adjusts the parameters of a descriptive model. The term is usually used in the context of Kohonen Maps and other, more confidential, unsupervised Neural Networks.

Supervised Training algorithms are based on the iterative minimization of a prediction error, but Unsupervised Training algorithms usually lack this nice mathematical framework : an unsupervised training algorithm usually does not try explicitely to minimize anything (although, sometimes it does, in fact, minimize a meaningful quantity, as is the case in k-means).
 

Generally speaking, Unsupervised Training algorithms are heuristic in nature : they work reasonably well most of the time, but they are not backed by a well understood theory. 

Typology

Typology is synonymous with clustering.

The word is very much in use in marketing circles. Typology, as the name implies, is used to identify a few "typical" customer profiles and behaviors, each being representative of a large number of observations that behave in similar fashions ("classes"). By partitioning a customer bases in homogenous classes, promotion actions can be targeted at particular classes with more efficiency than would be possible with an all-out promotional effort. 

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