A "vector" in vector search solutions is a dense vector generated by a transformer model. A sentence like 'a fat cat sat on a mat and ate a fat rat' put through a model becomes an array of floating-point numbers like [0.183, -0.774, ...], with hundreds of values (often 768). The point is that this vector is positioned in a 768-dimensional space in close proximity to semantically similar sentences. So then searching by semantic meaning is a simple (well...) exercise in measuring the distance between your query and the surrounding vectors.
We have a whole course on the topic, coincidentally also on the front page of HN today: https://www.pinecone.io/learn/nlp