What is vector search and how does it work in AI applications?

I'm learning about AI, large language models (LLMs), and Retrieval-Augmented Generation (RAG), and I frequently come across the term vector search. I understand it involves searching using vector embeddings instead of exact keyword matching, but I'm not sure how it works in practice.

Can someone explain what vector search is, how it uses embeddings to find semantically similar data, and why it's important for applications like RAG, semantic search, recommendation systems, and AI chatbots? A simple example using a vector database such as FAISS, ChromaDB, or Pinecone would be very helpful.