Fraud, equipment failure, network intrusions — the interesting class is often so rare that a supervised classifier never sees enough positive examples to learn from. Anomaly detection reframes the problem entirely: instead of learning what the rare class looks like, learn what normal looks like, and flag anything that deviates.
k-means needs to be told $k$ in advance, and assumes every cluster is roughly round. This page covers two families that relax those assumptions in different directions: hierarchical clustering defers the choice of cluster count to a visual cut, and density-based clustering abandons the "round cluster" assumption entirely by defining a cluster as a connected region of high density.
k-means is the clustering algorithm everyone reaches for first — and its simplicity conceals just how strong its assumptions are. It assumes clusters are round, similarly sized, and similarly dense, and it will happily produce a confident, wrong answer when those assumptions don't hold.
Before picking an algorithm, answer one question: what does the training signal look like? A dataset of (input, correct-output) pairs calls for a different family of methods than a pile of unlabelled data, which in turn differs from a system that only gets a delayed reward for a sequence of actions. The paradigm is chosen by the data you have, not by which algorithm sounds most impressive.
Principal component analysis answers a simple question with surprisingly deep machinery: which few directions in a high-dimensional dataset capture most of what's actually going on? The answer — rotate onto the axes of greatest variance — turns out to be one of the most reused mathematical results in all of machine learning.