In Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?
Think of a game like pool. It consists of the following items - a pool table, 15 different colored balls, 1 white ball and a cue stick. Let us say that you don't know how to play pool. Now, I buy all the items mentioned above and give them to you, can you play the game of pool? You obviously can't because you don't know how these items are supposed to interact with each other. So, the next step would be that I explain to you how these items interact with each other and lay down some ground rules for a game of pool.
So, to learn the game of pool, I gave you two things - (collection of items) + (interaction explanation & rules).
A space is something like the game of pool. The set involved is akin to the collection of items above and the "structure" is akin to the explanation of interaction between the various elements of the set and some ground rules on those interactions.
There are different kinds of spaces, therefore different kinds of structures. There is no such thing as "space" in general. You may consider a topological space, with a structure of open sets (fulfilling certain properties) - like a plane with open balls, its sums and so on. You may consider an algebraic object, like vector space, with algebraic structure, given by, for example, addition. You may actually have different structures on one space, and what you study are the interactions between them: the Euclidean plane has a both topological and algebraic structure, and these structures interact.
There is a formal definition of structure, see here. In this definition, structure is clearly distinguished from property and from something else called stuff. This formalism may look highbrow, but on closer inspection it's not that complicated and makes a lot of sense.