they all are dual-core (cpu's) but dual-core are now old and have weak performance as compared to core 2 duo, C2D are trademark while dual-core aren't personally core 2 duo are upgraded version f dual-core which have more cache memory high frequency against dual-core even C2D have good overclocking.and i3 are now the latest version of dual-core cpu's as i mention in the first line they all have 2 cpu's but i3 have 4 threads while dual-core and C2D have 2 threads.so i3 is king of dual-core cpu's.
dual are the models like these E2160,E2180,E2200 which have low frequency and low FSB speed like upto 800mhz.
core 2 duo registered trademark are he models like these E5200,E6300,E8400 which have higher frequency and high FSB speed like 1066mhz.
core i3 are the models like these i3 530,540,550,560 (these are 1st gen models) they are faster than both dual-core + core 2 duo
2nd gen which are named sandy bridge like i3 2100 which are much strongest cpu highest frequency 3.10ghz 2100 model has integrated intel HD 2000 graphics which are pretty faster than some external video cards while the other haven't any graphics.
dual core lithography are which also known as nm (nonometer) are 65
core 2 duo are 45 nm
i3 2100 has latest 32nm.