Bayesian Inference is a way of combining information from data with things we think we already know. For example, if we wanted to get an estimate of the mean height of people, we could use our prior knowledge that people are generally between 5 and 6 feet tall to inform the results from the data we collect. If our prior is informative and we don't have much data, this will help us to get a better estimate. If we have a lot of data, even if the prior is wrong (say, our population is NBA players), the prior won't change the estimate much. You might say that including such "subjective" information in a statistical model isn't right, but there's subjectivity in the selection of any statistical model. Bayesian Inference makes that subjectivity explicit.
Bayesian Inference can seem complicated, but as Brandon Rohrer explains, it's based on straighforward principles of conditional probability. Watch his video below for an elegant explanation of the basics.
If you'd like to try out some Bayesian statistics yourself, R has many packages for Bayesian Inference.
Data Science and Robots Blog: How Bayesian inference works