Statistical inference is the process of drawing conclusions about a population on the basis of measurements made on sample taken from that population. A typical example of statistical inference can be found in election polls. Before an election, survey companies ask groups of usually a few hundred to a few thousand people how they plan to vote, this group of people who are surveyed constituting the sample. On the basis of the data they obtain from the sample, the company then makes a prediction about the population, which consists of all eligible voters (typically many millions). It can be shown mathematically that, providing the sample is collected properly and is truly representative of the population, then it is quite possible to make very accurate inferences about populations consisting of many millions on the basis of samples of only a few thousand or less. Of course, in practise it is often very difficult to find samples that are truly representative of the underlying population of interest, leading to many complications in the process of making statistical inferences.
Statistical inference: A very clear and concise introduction to the purpose and process of statistical inference
Statistical inference and estimation: Discussion of key concepts with some examples