Degrees of freedom are an integral part of inferential statistical analyses, which estimate or make inferences about population parameters based on sample data. In a calculation, degrees of freedom is the number of values which are free to vary. As an illustration, think of people filling up a 30-seat classroom. The first 29 people have a choice of where they sit, but the 30th person to enter can only sit in the one remaining seat. Similarly, if you calculated the mean of a sample of 30 numbers, the first 29 are free to vary but 30th number would be determined as the value needed to achieve the given sample mean. Therefore, when estimating the mean of a single population, the degrees of freedom is 29.
Degrees of freedom are important for finding critical cutoff values for inferential statistical tests. Depending on the type of the analysis you run, degrees of freedom typically (but not always) relate the size of the sample. Because higher degrees of freedom generally mean larger sample sizes, a higher degree of freedom means more power to reject a false null hypothesis and find a significant result.