The linked article explains these in details.

I'll try to give a description - please correct any errors:

Large language models encode information about tokens (eg strings) as a large vector - eg 2048 values between 0 and 1 (as 16 bit floats).

If you give an LLM a document, it will produce a vector that one might think of as characterizing the "meaning"/"content" of the document, in the context of the LLM. This is equivalent to indexing the document.

Index many documents and store the vectors; then "index" a search query - and return the document vectors that are nearest to the query vector.

This way you might get a pointer to Rome and Juliet and various sonnets if you search a database of Shakespeare for "love stories".

A vector database is a database that makes it easy and quick to store, index, search and retrieve vectors.