A list of numbers treated as a single thing. Position on a map is a vector of two numbers; a word’s meaning inside a language model is a vector of a few thousand. Once something is a vector you can measure distance and direction, which is how machines compare things that are not obviously comparable.
Technical view
An element of a vector space, supporting addition and scalar multiplication. In machine learning, effectively always a fixed-length array of floating point numbers representing a point in a high-dimensional space. Direction usually carries the meaning and magnitude often does not, which is why cosine similarity is preferred to Euclidean distance for embeddings, and why normalising to unit length is routine.
Related
Part of AI Terminology Index, section: Maths and Statistics.