Core inflation is the inflation rate that excludes items with highly volatile prices.
Price indices are used to calculate the inflation rate (see Calculate Inflation) and one of the most common ways to do this is by using the Consumer Price Index (CPI). This index compares the cost of specific goods over yearly periods and consists of many items including food and energy. As you've probably noticed, energy prices tend to fluctuate quite wildly and this can have the effect of causing the inflation rate to fluctuate as well. So along with the normal inflation rate you always hear about, a Core Inflation rate is also calculated that tries to remove those items that tend to have prices that change significantly and sporadically over time.
We should note that if Core Inflation is used to describe the inflation rate that excludes certain items, there is also a term that is used to describe the inflation rate that includes all items, regardless of their volatility; it is called Headline inflation.