Definition
Statistical inference is the discipline of using sample data to draw conclusions about the underlying probability model or the population from which the data came. It is the inverse of probability: probability reasons from a known model to predict data; inference reasons from observed data back to plausible models.
The main inferential operations are estimation (best-guess of a parameter, plus its uncertainty), hypothesis testing (does the data refute a hypothesised value?), and prediction (what will future observations look like?).
Why it matters
How it works
Inference starts with a model: a family of probability distributions indexed by unknown parameters. Given data, you ask which parameter values are consistent with what you observe. A frequentist answers by computing point estimates (e.g. the maximum likelihood estimate), confidence intervals, and p-values. A Bayesian computes a posterior distribution over parameters, summarising what is still uncertain after seeing the data.
The classical workflow — collect data, fit a model, report estimates and uncertainty — is the backbone of empirical science. The rigour with which it is done determines whether the conclusions are trustworthy.