Core 2 Duo is based on LGA775 (a type of socket), and uses the old NB/SB connections to other peripherals like the graphics card and memory. This introduces latency ("lag") and slowdowns to the connections. It is a very popular Intel product, and is composed of a line of processors ranging from budget to enthusiast.
Core i3 is part of a newer series of processors using LGA1156. It departs from the NB/SB configuration and instead uses a direct media interface (DMI) to communicate with RAM and PCI-e components. This eliminates a lot of latency and bottlenecks, and allows faster communication and better performance. These are usually your video cards. It is much faster and more energy efficient. It is based on the 32nm manufacturing process.
Although they are both dual cores, the i3 is a better choice for an everyday system. Aside from being energy-efficient, it is also faster and supports newer technologies like DDR3, which is almost at the same price as DDR2 is now.
I'm sure there is a more detailed explanation, but this should give you a general idea of their difference.