Formally, black holes are a prediction of Einstein's theory of gravity. They have been observationally confirmed.
Regarding theory: The field equations of Einsteinian gravity, the Einstein Field Equations, admit solutions. The first closed form solution of these equations was found in 1916 by Karl Schwarzschild which is the massive non-rotating, uncharged black hole with an event horizon. Then Reissner, Nordstrom, Weyl, and Jeffrey found the case of the non-rotating and charged black hole soon after. Compared to the Schwarszchild black hole which has its event horizon at the Schwarszchild radius, the Reissner-Nordstrom metric has two horizons (one event horizon and one Cauchy surface), the locations of which depend on the Schw radius and on the electric charge of the black hole. If the electric charge is equal or greater to the mass of the black hole (in units where G=c=1), then the black hole may form in nature but would be a naked singularity since the horizons no longer cover the physical singularity.
It is an electric charge we are talking about, right? Outside of the event horizon, correct?
Yes, electrostatic charge. The Reissner-Nordstrom metric assumes that all the mass and electric charge reside at the physical singularity of the black hole:
The assumption leads to the prediction that light from the outside universe concentrates infinitely at the inner horizon, which contradicts the assumption that the black hole is empty except at its singularity. In reality, if there were such a thing as a charged black hole, as you approached very close to its inner horizon, you would see a rapidly growing explosion of light from the outside universe. The light triggers the mass inflation instability. From the inner horizon on, the Reissner-Nordström geometry is not physically realistic, despite being an exact mathematical solution to Einstein's equations.
Several decades later, Roy Kerr discovered the solution for the general case of an uncharged, rotating black hole in the 1960s. Lastly, the solution for the charged, rotating black hole is called the Newman-Kerr black hole: so this one has mass, spin, and electric charge. See the table of types of black holes here.
It is not expected that black holes will form in nature with significant electric charge because electromagnetic repulsion in compressing an electrically charged mass is dramatically greater than the gravitational attraction (by about 40 orders of magnitude). Also, black holes can accrete material from their surroundings, and thus gain negative or positive charge, however the effect of accretion on the black hole's NET charge is negligible.
Regarding observation: Astrophysically, black holes are expected to have negligible charge, except in certain circumstances involving charged particle phenomena near black holes, e.g. cosmic rays, but they are expected to have spin! And we've observed lots of black holes now, stellar mass black holes in X-ray binaries and in binary merger events, and supermassive black holes in galactic nuclei.
It is possible that all of the black holes that we've discovered thus far DO have electric charge, but that it is so small that its very difficult for us to measure currently, so we don't know if they have charge or not. Future observations are being explored for trying to measure the electric charge of a black hole, for example this framework uses retro-lensing.
Concerning observations of the astrophysical formation of black holes, all observations to date indicate that the Kerr metric is the correct description of black holes that form in nature. This is why we currently consider the no-hair theorem to be valid, along with the astrophysical expectation of negligible electric charge, although it is still unproven in full generality. Future observations of black hole electric charge could force us to update this picture, however! IF that happens, then the Kerr-Newman metric would replace the Kerr metric as the best description we have for black holes in nature.