What are vectors?

isn't a vector what Sulu puts into the computer before going to the 'Neutral Zone in Star Trek?

Hi Pamela,
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Vectors are the mathematically described points that make up the structure (skeleton) of vector art. These points can be anchor points (there are four types of anchor points: corner, bezier curve, Combination Cusp, Reflex Curve Cusp) as well as mesh points (which are slightly different in that they have up to 4 path segments radiating out from each point and can also contain color information). The paths are the "lines" that connect these points and provide the structure for your art. Paths are defined by anchor points and direction handles.
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Beginners often confuse paths (part of the skeleton structure) with strokes (appearance attributes that appear as outlines). One path can have as many strokes as you prefer. The old Barnum and Bailey Circus Logo is an example of a path with many different strokes applied to the same path.
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Note that this vector structure (skeleton) of points and paths is only visible inside a vector drawing program like Adobe Illustrator... it does not show (it is invisible) in print or screen when you publish. When you publish you are actually seeing the Appearance attributes (The fills, strokes, transparencies, effects, etc.).
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Finally, vector programs come in two categories
1. Object Vector programs like Adobe Illustrator or the vector tools in Adobe Photoshop and
2. Shape (Merge) Vector programs like Adobe Animate (which also has the ability to switch to Object vectors at any time).
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For example. a square is described by 4 corner anchor points connected by 4 path segments composing a path. A circle is described by 2 bezier curve points connected by 2 path segments composing a path. If you would like to see your artwork outside of Adobe Illustrator, then you must apply strokes, fills and other appearance attributes. Without the strokes, fills, etc. your artwork will be invisible.
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Advantages: Because the vectors are mathematically described (behind the scenes in the program), your artwork can be resized to any size (for example, as small as an amoeba or big enough to cover a billboard) without any loss in quality. That is what is meant by the term "resolution independent." The art always remains crisp and clean. Even better, the mathematics behind all this allows for non-destructive editing. Anything you do can be undone (even after you save your document) at any time.
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Hope that helps.

Good question. Thank you for asking.

Typically we think of Photoshop as producing pixel-based (sometimes call bitmap) art and Illustrator as being a vector art based app.

However, Photoshop added excellent pen tool vector art option back in the CS6 days.

We like to think of vector graphics in terms of visuals results of the paths they create but the underlying technology is relative to the  the x- and y-axes those points create on a document.

Does that make sense?