Short answers, pulled from the story.
Statistical classification is the process of assigning a new observation to one of a set of predefined categories based on patterns learned from past data. Algorithms analyze each observation as a feature vector of measurable properties and use statistical methods to determine the most likely category. The field covers both binary problems with two classes and multiclass problems with three or more.
Fisher carried out early foundational work on statistical classification, focusing on the two-group problem. His contribution was Fisher's linear discriminant function, a rule for assigning a new observation to one of two groups. His early work assumed that data values within each group followed a multivariate normal distribution.
Frequentist classifiers use only the observed data to assign categories, with no assumption about how common each group is in the broader population. Bayesian classification procedures incorporate prior information about the relative sizes of different groups, giving them a natural advantage when base rates matter. Bayesian procedures tend to be computationally expensive, and approximations were needed before Markov chain Monte Carlo computations were developed.
A probabilistic classifier outputs the probability of an observation belonging to each possible class, rather than simply naming a single best class. This allows the classifier to report a confidence value and to abstain from predicting when confidence is too low. Probabilistic classifiers also reduce error propagation when integrated into larger machine learning pipelines.
A feature vector is a structured collection of measurable properties describing a single observation. Features can be binary, categorical, ordinal, integer-valued, or real-valued. For an image, feature values may correspond to individual pixels; for a text document, they may represent the occurrence frequencies of different words.
Statistical classification is applied in medical image analysis, drug discovery, microarray analysis, internet filtering, and identification tasks. It is used both as a data mining procedure and as part of more detailed statistical modeling. The field has produced a large toolkit of algorithms because no single method is appropriate for all data sets.