A Gaussian process is a probabilistic model that treats an unknown function as a random object and places a distribution over all the functions it might be. Rather than fitting a fixed equation with a handful of parameters, it represents beliefs about the function’s behavior everywhere at once. The standard reference, Rasmussen and Williams’ freely available 2006 book, defines it as a collection of random variables, any finite subset of which has a joint Gaussian distribution.
In practice you specify a covariance function, often called a kernel, that encodes assumptions about how the function should behave: how smooth it is, how quickly it changes, and how strongly nearby inputs are correlated. Given some observed data points, the Gaussian process produces a predictive distribution at any new input, returning both a best estimate and a measure of how uncertain that estimate is.
The defining strength is this built-in uncertainty. Where data are plentiful the predictions are confident with narrow error bars, and where data are sparse the error bars widen automatically, so the model is honest about extrapolation.
That property is why Gaussian processes are valued in settings where the cost of being wrong is high or where each new observation is expensive to obtain. They are central to Bayesian optimization, used for tuning systems and designing experiments, because they help decide where to sample next by balancing what looks promising against what remains unknown.