Curvature of space (i.e., a curved space) is space that deviates from the well-known rules of geometry. Curved spaces can be consistently described mathematically, and general relativity imagines the space of the universe as non-flat, to help describe gravity, and cosmological theories assume some curvature. In both cases, the curvature is too small to show up in day-to-day measurements.
A curvature of space can be detected by testing the known rules of flat geometry, in an analogous manner to how one might measure the curvature of the Earth's surface through measurement: for example, laying out a large triangle on the surface consisting of a given Earth altitude (e.g., sea level) will yield a shape whose angles do not add up to 180° or laying out a circle and measuring the ratio of its circumference and diameter will not yield π. Precisely these same tests can, in principle, be done in space, and given enough curvature and large enough shapes, the curvature could be measured.
A curvature of space can be positive (analogous to the surface of a sphere) or negative (analogous to the surface of a saddle shape, or the inner edge of a doughnut). Sensitive trials have been carried out to see if any curvature is detectable.