A vector is something that has both Magnitude (size, length, whatever) and Direction (which ways is it pointing). A vector is not a shape, for example an arrow as it is commonly drawn. There are many ways of expressing vectors, but that's all math mumbo jumbo.

Vectors are used when you want to show that something has a directed value. If I were to measure the temperature all throughout a room, I would not need vectors because I'd just need to know the temperature at all points (that is, temperature is a Scalar because it only has magnitude). However, if I instead wanted to know the air currents in the room I'd probably ask "If I were to leave some incredibly light thing at this point, which way would it be pushed by the air, and how hard would it be pushed". I could express this as a vector at that point - the strength of the air current there along with its direction. Only giving one or the other is not helpful here, but both let us paint a picture of the room and how anything will move there.

Vectors aren't always straight lines though. For example, if I wanted to know how a bicycle wheel turned I'd need to know both the speed (ie magnitude) and whether it was clockwise or counterclockwise (direction). In this case, the only allowable directions are clockwise or counterclockwise, so you only need 1 variable to express this because this is a one dimensional system. In 3 dimensions you need 3 variables, eg North vs South, East vs West, and Up vs Down.

Sometimes people use vectors to show a location, when they actually only care about the endpoint. This is not technically a correct physical interpretation of what a vector is, but it allows us to do fancy math mumbo jumbo to get useful answers so we give it a pass.