What is the difference between Intel Core i3 ,Core i5 and Core i7?

Core i3 belongs to a family of dual core, 64-bit, x86 CPUs from Intel intended for entry-level desktop and laptop computers. Introduced in 2010, the Core i3 is the third line in Intel's Core "i" branding (i3, i5, i7). The Core i3 chips do not include the Turbo Boost feature, which is found in i5 and i7 chips but include Hyper threading(that allows each processor core to work on two tasks at the same time for greater multitasking)technology.

Core i3 processors have an integrated GPU and two cores (all generations including first, second and third are dual core processors)

Core i5 belongs to a family of dual (first generation) and quad (second and third generation) core, 64-bit, x86 CPUs from Intel, introduced in 2009. The Core i5 chips are the midrange CPUs in the Core "i" line between the entry-level i3 and the high-performance i7 series. Certain models of the i5 family include the Turbo Boost feature

A family of 64-bit x86 CPUs having quad cores from Intel that were introduced in 2008 as the successor to Core 2.  The first models included a graphics processing unit (GPU) in the same chip package as the CPU including vastly improved graphics, battery life, and security—for a zero-compromise computing experience. With key features such as  Hyper-Threading Technology,(that allows each processor core to work on two tasks at the same time for greater multitasking) to the Intel® Iris™ graphics for stunning 3D visuals and faster, more advanced video and photo editing, the 4th generation Intel® Core™ i7 processor delivers top-of-the-line performance for your most demanding tasks.

• Enhanced Memory Control

supporting three channels of DDR3 memory, the Core i7 contains its own memory controller that runs with a separate clock in the "uncore" area (a part of the chip that is not executing program instructions).

• Quick Path and Turbo Boost

a major feature is the Quick Path system interconnect, which is a high-speed, packet-oriented, point-to-point pathway between the CPU cores and memory and between the cores and I/O hub. Depending on the multiprocessor capability supported by the chip model, an integrated crossbar router between the cores may be included.

The i7 line also introduced Turbo Boost, also available in some i5 models, which allows the speed of CPU cores to be adjusted 

In Short,

Core i3, i5 first gen= 2 core
Core i3 2nd and 3rd gen=2 core
Core i5 2nd and 3rd gen= 4core
core i7 =4 cores