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The Intel Core i3 processor is the closest successor to the now out-of-production Core2Duo processor. The most significant differences between the i3 and i5/i7 is the lack of Turbo Boost and less cache (L3) memory. The i3 offers moderate all-around performance and is often found in budget-oriented systems.
A processor core is a processing unit which reads in instructions to perform specific actions. Instructions are chained together so that, when run in real time, they make up your computer experience. Literally everything you do on your computer has to be processed by your processor.