A very simple vector example:

Assume we have an array of 26 binary variables, one for each letter of the alphabet. We flip to 1 if the letter is present in the word. Dog -> [0,0,0,1,0…] Cat -> [1,0,1,0,0…]

Then we can do a search by taking a target word and doing a cosine similarly search, roughly we find the binary array most similar to a given array to compute a score.

Modern models compute a much more complicated vector by using the context of the words around them. Multimodal image models use a combination of images and text to train a model so later you can use pass it some text to get a vector.

There are some things out there about doing math on vectors because the form a latent space, for example: ‘king - man + woman = queen’ but empirical tests show that this doesn’t quite hold up on modern models. 3blue1brown has a video on vector math which is worth a watch.