an open set
Think of a region with no fence at its edge. If you stand anywhere inside an open set, you can always take at least a tiny step in every direction and still be inside it — there is no point of the set sitting right on a boundary with nothing-to-spare. The open interval (0, 1) on the number line is the model: any point in it, say 0.999, has a little room around it still inside (0, 1), but the endpoint 1 is deliberately left out, because at 1 you would be standing on the brink with no wiggle room.
On the ordinary line or plane the precise test uses small balls. A set U is open if, around every point p in U, there is some radius r > 0 so that the whole ball of radius r centred at p stays inside U — every point has elbow room. In a general topological space there are no radii, so the logic is flipped: the open sets are simply the family you declared at the start (subject to the three axioms), and a set is open precisely when it belongs to that family. A neighbourhood of a point p is then any set that contains an open set containing p; that is the topological version of 'a region surrounding p'.
Open sets are the load-bearing idea of all topology: continuity, limits, connectedness, and compactness are every one defined purely in their language, with distance nowhere mentioned. The honest subtlety is that 'open' is not the opposite of 'closed' in everyday speech. A set can be both open and closed (like the whole space, or the empty set), or neither (like the half-open interval [0, 1) on the line). 'Closed' means the complement is open — it does not mean 'not open'.
On the plane, the disk of all points with x^2 + y^2 < 1 is open: any point inside has distance less than 1 from the centre, so a small circle around it still fits inside. But the disk x^2 + y^2 <= 1 is not open, because a point exactly on the rim (where x^2 + y^2 = 1) has no room — every ball around it pokes outside.
Open means strict inequality, no boundary included; the rim is exactly what openness excludes.
Do not read 'open' and 'closed' as opposites: a set may be both (clopen) or neither, and 'closed' is defined as 'complement is open', not as 'fails to be open'.