A black hole is a region of spacetime from which nothing can escape. More formally, the future light cone of any observer within the black hole is completely contained in the black hole, and the black hole region is not within the past light cone of any observer that goes to spatial infinity in an infinite amount of time.
Brief Summary
Intuitively, black holes are compact regions of spacetime from which nothing, not even light, can escape. They were originally conceived within newtonian-gravity by considering a body so massive that the escape velocity at its surface would be larger than the speed of light (and hence they would be seen as black), but their study became much more profound within the framework of general-relativity.
Black Holes in General Relativity
The first black hole solution of the Einstein Field Equations, the main equations of General Relativity, was the Schwarzschild solution, which describes a spherically symmetric, vacuum spacetime. By comparing the solution to the predictions of Newtonian gravity, one finds out that it is characterized by a parameter $M$ which can be identified as the mass of a spherical body at the center of the spherical system of coordinates. Spacetime is then described by the metric $$\textrm{d}s^2 = - \left(1 - \frac{2M}{r}\right)\textrm{d}t^2 + \left(1 - \frac{2M}{r}\right)^{-1}\textrm{d}r^2 + r^2 \textrm{d}\theta^2 + r^2 \sin^2\theta \textrm{d}\phi^2,$$ where geometrical units, with $G = c = 1$, are employed.
One can then notice the presence of two disturbing features of this metric-tensor: it is ill-defined at $r = 2M$ and at $r = 0$. While the former was eventually seen to be a coordinate singularity, just like the issues one has at $\theta = 0$ and $\theta = \pi$ due to the use of spherical coordinates, the latter was deemed problematic and possibly unphysical.
The reason for the belief that the solution could be unphysical is because the high degree of symmetry could indeed conduct the appearance of singularities, as it already did in Newtonian gravity. For example, a spherically symmetric cloud of dust in Newtonian gravity will collapse at the origin and form a singularity due to the matter achieving infinity density at that point. However, if the cloud was rotating in the slightest bit, the centrifugal effect would keep the singularity from forming. Hence, it was not clear whether the singularity on the Schwarzschild solution was a mere accident due to symmetry or an actual prediction of General Relativity.
While the Schwarzschild solution was proposed in the 1910s, the matter would only be settled in the 1960s, with the work of R. Penrose and S. Hawking concerning the singularity theorems, which employed global techniques to show that the formation of singularities is indeed a robust prediction of General Relativity and not a mere consequence of symmetry considerations. Penrose would later be awarded half of the 2020 Nobel Prize in Physics due to his work on this.
There are more black hole solutions within General Relativity. Usually, one considers the stationary solutions (i.e., those that possess a time translation symmetry). If one assumes all matter on spacetime is comprised of electromagnetic fields, the no-hair theorems implies that the only possible solutions are the reissner-nordstrom-metric, the kerr-metric, and the kerr-newman-metric (the latter having the remaining ones as special cases).
Astrophysical Considerations
Black holes can be formed by means of the gravitational collapse of a sufficiently massive star when it can continue interacting gravitationally with its surroundings. For example, gravitational-wave-detectors, such as LIGO, Virgo, and KAGRA, have observed a number of black hole and neutron star mergers. Notice then that even though black holes do not emit light, they can be studied experimentally by means of their gravitational influence.
While this approach involved gravitational-waves, there are also other possibilities. Analyzing the orbits of stars at the center of the Milky Way led to the discovery of the presence of a "supermassive compact object at the center of our galaxy." In the words, chosen by the Nobel Prize when justifying half of the 2020 Nobel Prize in Physics being awarded to Andrea Ghez and Reinhard Genzel for this discovery.
Furthermore, the presence of matter surrounding the black hole and the gravitational-lensing effects due to the black hole's massive gravity also provide a way of performing experimental observations. By exploiting this, the [Event Horizon Telescope] (https://en.wikipedia.org/wiki/Event_Horizon_Telescope) was able to take the first picture of a black hole.
