Curved space is defined to be anywhere where the rules of geometry
'do not work'. For example, the angles of a triangle add up to more than 180°.
It is easier to explain this using two-dimensional examples because we can look into it from the outside. Consider a 2D plane. The internal angles of a triangle drawn on that plane will always sum to 180°. Now consider the surface of a sphere. If you draw a polygon with three straight sides, the internal angles will sum to greater than 180°. In fact, the excess angle is proportional to the area of the triangle.
To give an example, imagine starting at the North Pole and drawing a straight line to the Equator, which has length x. Turn 90° and draw a line of length x along the equator. Turn back towards the North Pole and draw another line of length x. This will bring you back to your starting point. You have drawn a polygon with three straight sides, hence it is a triangle, but the internal angles add up to 270°.
When this excess angle is greater than zero, the space is said to postively curved. When it is less than zero, space is negatively curved.
Another way to measure the curvature of a space is to measure the circumference of a circle and see how it compares with the circumference predicted by the formula 2 PI * radius. If we draw a circle around the North Pole at a distance of x, the circumference will be less than expected since the radius of the circle is actually the perpendicular distance between the circle and the central axis of the Earth, not x.
What about in three-dimensional space? This is harder to comprehend because the human mind has trouble imagining things outside of three dimensions. We can measure how 3D space is curved by using the circle example above. If the circumference is less than expected, the space is positively curved, if it is greater, space is negatively curved.
"But circles are 2D", I hear you cry, "aren't we dealing with 3D?". That is true and curvature of 3D space is orientation dependent. Changing the orientation of the circle will give different readings for the curvature of the space. It would seem logical to just use a sphere and see if the surface area is greater than or less than expected. This does indeed work but will give an average curvature for that region of space.