A vector database stores vectors of fixed dimensions (called embeddings) such that we can then query the database to find all the embeddings that are closest (most similar) to a given query vector using a distance metric, which is usually the cosine similarity, but we can also use the Euclidean distance.
The database uses a variant of the KNN (K Nearest Neighbor) algorithm or another similarity search algorithm.
Vector DBs are also used for finding similar songs (e.g. Spotify), images (e.g. Google Images) or products (e.g. Amazon).