i types are the new name for Intel CPU's. All future CPU's will be branded either i3, i5, or i7 types, as the Intel Core 2 DUO line is phased out, to make it easier for users to choose between models.
Basically i3 is the entry level, i5 is the mid consumer level, and i7 is for the pro level consumer market
The i3 is indeed a dual core CPU, but has a lower clock speed and configurations than a i5 would have. For example, you will not get Turbo Boosting with i3, which is basically where it uses tricks to get the type of speed as a Quad on a Dual core within some programs that are capable of using it (though of course, not as capable or fast)
The i3 is the new line, and will be just as fast as the Intel Core 2 duo capability wise, though of course, depending on the different clock speeds, may be slightly slower or faster than that branch of processors.
The i series I believe also has a built in GPU (graphics processor) within the processor itself, and with the power requirements of these new processors being pretty low, you can expect a much higher battery life on a laptop using them