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Dimensions
Vectors are lists of numbers that represent coordinates in space... cool. 2D space is easy: "3 units right, 2 units up" is [3, 2]. 3D is also pretty easy; we just add a z coordinate: "3 units right, 2 units up, 1 unit forward" is [3, 2, 1].
The problem is that two or three dimensions isn't nearly enough to represent much semantic meaning. Semantic embeddings often have 300+ dimensions.
Yes, I know there aren't 300 dimensions (that we know of?!) in the physical universe, but in the mathematical world of vectors, we can have as many dimensions as we want. It's still just a list of numbers.
Example
"The Great Bear" → [0.2, -0.8, 0.1, 0.7, -0.3, ...]
"Ted" → [0.1, -0.9, 0.2, 0.6, -0.2, ...]
"Backcountry" → [0.8, 0.1, -0.5, 0.2, 0.9, ...]
These vectors might have 300+ dimensions, but the key idea is the same: distance in this space represents semantic similarity.
Visualizing 300 dimensions is basically impossible. Geoffrey Hinton, often considered the "Godfather of AI," suggested that the way to visualize 14-dimensional vectors is to visualize a 3D space and say "Fourteen!" to yourself very loudly... and we're dealing with way more than 14 dimensions here.