Picture one: an arrow
Imagine standing at a corner and telling a friend, 'walk 4 steps east and 1 step north.' That instruction is a vector: it has a length (how far) and a direction (which way). Draw it as an arrow from where you start to where you end up.
Picture two: a list of numbers
We can write that same instruction as a short list: (4, 1). The first number is the east amount, the second is the north amount. These are its coordinates — the numbers that pin the arrow down exactly.
arrow: 4 east, 1 north list: (4, 1) same vector, two notations
Flipping between the two pictures is the single most useful habit in linear algebra. Arrows give you intuition; lists let you compute. A scalar like 2 simply doubles every number in the list.
Beyond the page: 3D and higher
Add a third number and you describe arrows in 3D space: (4, 1, 2) might mean east, north, and up. There is no reason to stop. A list of 100 numbers is a vector in 100 dimensions — you can't draw it, but you can still add and scale it exactly the same way.