Vector
A list of numbers that stands for a thing — and, read as coordinates, a direction in space.
Like you're twelve
Describe a song with five numbers: how fast, how loud, how sad, how much guitar, how danceable. Two songs with nearly the same five numbers are nearly the same song, and you can tell that without listening to either. You did not compare the songs; you compared their lists.
Now use two thousand numbers instead of five and the same trick describes anything — a face, a sentence, a customer, an hour of sensor readings. That list is a vector.
Precisely
A vector is an ordered array of numbers of fixed width. That width is the dimensionality of the space, and in HeatherDB it is fixed per database: every vector in one database has the same number of components, because comparing lists of different lengths is meaningless.
The numbers themselves come from wherever you like — an embedding model, a sensor, a hand-built encoding, or random assignment for symbols that only need to be distinguishable. Nothing about the store cares. What matters is that similar things get similar lists, because closeness is the only thing the memory can see.