What is a vector database?
A vector database stores embeddings and finds the ones nearest to a query embedding quickly. It is the index behind semantic search and retrieval-augmented generation. It can be a dedicated product or an extension of a regular database, such as pgvector for PostgreSQL, which keeps vectors next to the rest of your data.
Nearest-neighbour search over millions of vectors uses approximate indexes such as HNSW or IVF, trading a little accuracy for a lot of speed.
For a personal library of hundreds or thousands of papers, a single Postgres database with pgvector is plenty.