The Federal Reserve's mandate to maintain price stability requires that whenever significant inflation threatens it is supposed to hit the economy on the head with a brick: raise interest rates, and so discourage investment spending, lower capacity utilization, raise unemployment, and so create excess supply. The Federal Reserve would rather not do this unless it has no other option. If the rise in inflation is thought to be (a) transitory and thus (b) self-limiting, the Fed would prefer to let sleeping dogs lie rather than hit the economy on the head with a brick.
The Fed cannot, however, just say "we regard this rise in inflation as (a) transitory and thus (b) self-limiting, and so are going to let sleeping dogs lie." A Fed that does that quickly loses its credibility as an inflation-fighter, and a modern central bank with no inflation-fighting credibility is in a world of hurt.
However, when increases in inflation are confined to (i) energy and (ii) food prices, odds are that the increase is transitory and will be self-limiting. Hence the concept of "core inflation." If the Federal Reserve concludes that the current rise in inflation is transitory and self-limiting, it can point to the core inflation number as a principled excuse for not hitting the economy on the head with a brick.
The Fed's favorite measure is the "core" deflator for personal consumption expenditures (PCE) reported by the Bureau of Economic Analysis (in addition to the GDP deflator, the BEA calculates separate deflators for each component. Overall GDP statistics are quarterly, but the BEA provides monthly data on consumption). Here's how it looks (% changes from 1 year ago):
Clearly overall PCE inflation is more volatile than the "core" - taking out food and energy prices makes inflation appear more steady. Also, core inflation has mostly been lower than overall inflation - food and energy prices have been going up faster than the prices of consumption goods generally. That is, the inflation rate the Fed is responding to is less than the one we consumers are experiencing. (BEA data via FRED).