vector space
A vector space is the playground where vectors live: a whole collection of vectors that you are allowed to add together and scale by numbers, with the guarantee that you never fall out of the collection. Add any two vectors in the space and the result is still in the space; scale any vector by any number and the result is still in the space. That self-contained quality is called being 'closed' under addition and scalar multiplication, and it is the heart of the idea.
On top of being closed, the operations must behave the way arithmetic should: addition can be done in any order, there is a zero vector that changes nothing when added, every vector has a negative that cancels it, and scaling distributes over addition. These are the standard rules (the axioms), and they simply formalize 'addition and scaling work normally'. The familiar examples are R^2 (the flat plane), R^3 (3D space), and R^n (lists of n numbers).
What makes the concept powerful is how far it reaches beyond arrows. The set of all polynomials is a vector space — you can add two polynomials and scale one by a number. So is the set of all smooth functions, or all 2-by-2 matrices. Because they all obey the same rules, every theorem you prove about vector spaces applies to all of them at once, which is why linear algebra shows up everywhere from physics to data science.
R^n, all ordered lists of n real numbers, is the standard vector space; n = 2 is the plane, n = 3 is space.
Every vector space must contain the zero vector — scaling any vector by 0 lands on it, so closure forces it to be there.