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Black holes form when the most massive stars (>20 solar masses) collapse to densities so extreme that spacetime itself curves into a region from which not even light can escape—the event horizon. The Schwarzschild radius defines the event horizon's size; within it, spacetime curvature becomes the dominant feature of the gravitational interaction.
You know from stellar nucleosynthesis that massive stars fuse progressively heavier elements in their cores — hydrogen to helium, helium to carbon, and so on up to iron. Iron is the endpoint because fusing iron consumes energy rather than releasing it. When a star more massive than roughly 20 solar masses exhausts its nuclear fuel and builds up an iron core exceeding the Chandrasekhar limit (~1.4 solar masses), no known force can support the core against gravitational collapse. Electron degeneracy pressure fails, the core implodes in milliseconds, and if the resulting object is too massive even for neutron degeneracy pressure to halt the collapse (above roughly 2-3 solar masses for the remnant), the matter collapses without limit — forming a black hole.
The defining feature of a black hole is the event horizon, a boundary in spacetime beyond which the escape velocity exceeds the speed of light. For a non-rotating, uncharged black hole, the event horizon is a sphere with radius equal to the Schwarzschild radius: R_s = 2GM/c², where G is the gravitational constant, M is the mass, and c is the speed of light. For the Sun's mass, this works out to about 3 kilometers — the Sun is not massive enough to become a black hole, but this gives you a sense of the extraordinary density involved. The event horizon is not a physical surface; it is a causal boundary. An observer falling through it would notice nothing locally unusual at the moment of crossing, but they could never send a signal back out.
From the perspective of general relativity, what makes black holes so remarkable is that inside the event horizon, the roles of space and time effectively interchange. In normal spacetime, you can move freely in space but are inexorably carried forward in time. Inside the event horizon, the radial direction toward the center becomes timelike — moving toward the singularity (the point of formally infinite density at the center) is no longer a matter of spatial motion but of the passage of time itself. Just as you cannot avoid moving into the future outside a black hole, you cannot avoid moving toward the singularity once inside. This is why nothing escapes: it is not merely that the gravitational pull is strong, but that all future-directed paths lead inward.
Real astrophysical black holes are almost certainly rotating, described by the Kerr solution rather than the simpler Schwarzschild solution. Rotation drags spacetime around the black hole in a phenomenon called frame-dragging, creating a region outside the event horizon called the ergosphere where nothing can remain stationary relative to distant observers. The existence of the ergosphere has profound consequences: it enables energy extraction from the black hole's rotation (the Penrose process) and is intimately connected to the powerful jets observed in active galactic nuclei and some X-ray binaries. Despite their reputation as cosmic destroyers, black holes are among the most important engines in the universe, shaping the evolution of galaxies through the enormous energy released by matter falling toward them.