ASIC stands for Application Specific Integrated Circuit.
It is customized for specific purpose rather than general purpose.
It can be a chip for cell phone, networking, communication or even for a children toy.
An ASIC can be designed by either full custom approach or semi-custom approach.
In full custom approach each transistor is hand crafted. So it take a lot of time and not suited for most consumer market. Because time to market will be very high. On other hand semi-custom uses pre designed stranded cell library. So time to market is less.
SOC
SOC is System-on-Chip.
SOC may contain multiple processor cores, different type memories like RAM, ROM, EEPROM, DSP processor , different kind of interfaces like USB, Ethernet, CAN, UART, pheripherals like Osillator, PLL, ADC,ADC, PWM on a single chip.
So it is a complete system on a single chip. It is not necessary that a SOC contains all of these modules. It contains modules depending on requirement.
Further a SOC can be designed using ASIC approach.