Theory of operation for a software defined radio
There is a four part series "Software radio for the masses", linked from the ARRL, which explains all this basic information, including the different ways of doing software defined radio.

Once the signal is digitized, it's processed using DSP techniques. You can find more details about them in "Signals, Samples and Stuff: A DSP Tutorial" articles, also linked from above mentioned page.

Oscillators in SDR can be of any kind, from crystal, to PLL controlled VCOs, to direct digital synthesis (DDS), to numerically controlled oscillators (NCO, CORDIC) done entirely in software or FPGA.