Any space that is not completely flat is called curved space. The surface of a sphere is curved space, as is the surface of a saddle. A sphere is an example of positive curvature, meaning if a triangle is made with straight lines in curved space, the angles will add up to more than the normal 180 degrees. A saddle is an example of negative curved spaced. Gravity is cause by space curvature — mass curves space, which forces objects to pull together.
The Pythagorean Theorem is often used to check if space is flat or curved. This math formula uses the length of each side of a triangle instead of angles. If the lengths match what the theorem states, then the triangle is in flat space. If the lengths do not match exactly with the theorem, then the triangle is in curved space. Angles are difficult to measure over long distances, but measuring the sides, or perimeter, of a triangle can easily display the nature of the space.
Euclidean geometry is the study of shapes in flat space. It is based on a list of basic information, called axioms, and proves many math concepts like the Pythagorean Theorem. The axioms are often disproved, meaning they are shown to not always be true, in curved space, or non-Euclidean geometry. All triangles have 180 degrees in Euclidean geometry, which is easy to disprove in curved space by measuring each angle with a protractor.
Curved space plays an important role in modern astronomy. Gravity is considered the curved space surrounding a large body that causes smaller objects to orbit or collide with the large body. This was not discovered until Einstein published his Theory of General Relativity which first described gravity as curved space. Before this, astronomers calculated orbits inaccurately because space was treated as a three-dimensional Euclidean shape. Modern astronomers can calculate and predict much more with non-Euclidean space, like black holes and how galaxies move.
Even the father of physics, Isaac Newton, used Euclidean geometry. It was the only way to study shapes for over 2000 years. Then, in the late 19th century, the axiom that parallel lines never cross was disproved by Janos Bolyai. Einstein was able to understand non-Euclidean geometry and how it could be used to correctly predict the bizarre orbit of Mercury. The modern view is that true Euclidean shapes only exist in spaces far away from any gravitational body.