Variational inference is a method for approximating probability distributions that are too complex to compute exactly. In Bayesian modeling, the quantity you usually want is the posterior distribution over hidden variables given the data, but for most interesting models that posterior cannot be written down or integrated directly. Variational inference replaces the impossible integration with an optimization problem, as laid out in the 2016 review by David Blei, Alp Kucukelbir, and Jon McAuliffe.
The idea is to pick a family of simpler, tractable distributions and then search within that family for the member closest to the true posterior. Closeness is measured by the Kullback-Leibler divergence, an information-theoretic gap between two distributions. Because optimization is generally faster than the alternative of drawing many samples, variational inference scales to large datasets and complex models where sampling methods would be too slow.
That speed is the central trade-off. Markov chain Monte Carlo methods are asymptotically exact but can be slow; variational inference is fast but only approximate, since the answer is constrained to whatever simple family you chose. The review authors note that the method is powerful but still not fully understood theoretically.
For a general reader, variational inference matters because it made Bayesian reasoning practical at scale. It is the engine behind variational autoencoders and many large probabilistic models, letting systems quantify uncertainty over millions of data points in reasonable time rather than days.