Difference Between Cores and Processors

Yes, a multi-core processor is a single piece of hardware ("one processor") that provides several cores than can work concurrently.

The i5 is a single processor that provides either 2 or 4 physical cores depending on model (see here).

Note that some Intel processors (the i5 included) use hyperthreading, a system where a single processor has (for example) 2 physical cores, but will provide 4 logical cores - allowing the operating system to treat the processor as having more cores than it really does.

Yes, a "core" is just a processor that's placed on to the same integrated circuit with other processors. See the Wikipedia article for further info on multi-core processors.