How to read RCA output from an mbed?
By "RCA output" I assume they mean a composite video output.  In order to convert that into some sort of digital image that a microcontroller can use, you would need an extremely fast ADC, or simply a video decoder (i.e. one from Maxim IC, Analog Devices, TI) to tease out the image data.  Better yet, just get a camera with a digital output, as the digital-to-analog-to-digital path is a waste, as it's costly and performs worse.

NTSC scanlines are transmitted at 15.734 kHz with frequency content up to 4.2 MHz, impossible to handle with a standard microcontroller ADC which have sample rates of 1 Msps at the high end (NXP LPC1768 only has 200 ksps).  Furthermore, as Leon points out, your memory is extremely limited.  If you read just one QVGA (320 x 240) frame into memory at 8 bits/pixel, that's 76.8 kB, which exceeds the mbed's (NXP LPC1768's) on-board 64 kB RAM.