Quantum particles sometimes cooperate to develop a macroscopically ordered state with extraordinary properties. Superconductivity and Bose-Einstein condensation are examples of such cooperative phenomena where macroscopic order appears spontaneously. Here, we demonstrate that such an ordered state can also be obtained in an optically excited semiconductor quantum well in a high magnetic field. When we create a dense electron-hole plasma with an intense laser pulse, after a certain delay, an ultrashort burst of coherent radiation emerges. We interpret this striking phenomenon as a manifestation of superfluorescence (SF), in which a macroscopic polarization spontaneously builds up from an initially incoherent ensemble of excited quantum oscillators and then decays abruptly producing giant pulses of coherent radiation. SF has been observed in atomic gases, but the present work represents the first observation of SF in a solid-state setting. While there is an analogy between the recombination of electron-hole pairs and radiative transitions in atoms, there is no a priori reason for SF in semiconductors to be similar to atomic SF. This is a complex many-body system with a variety of ultrafast interactions, where the decoherence rates are at least 1,000 times faster than the radiative decay rate, an unusual situation totally unexplored in previous atomic SF studies. We show, nonetheless, that collective many-body coupling via a common radiation field does develop under certain conditions and leads to SF bursts. The solid-state realization of SF resulted in an unprecedented degree of controllability in the generation of SF, opening up opportunities for both fundamental many-body studies and device applications. We demonstrate that the intensity and delay time of SF bursts are fully tunable through an external magnetic field, temperature, and pump laser power.