Mic input and Speaker output using arduino
Microphone input is easy (ADC). Speaker output is easy (PWM). Both have many, many examples strewn across the Internet.

What happens in between is not, and will require external components. You will need at the very least somewhere to store the samples between record and playback. Investigate SPI SRAM large enough to store all the required samples at your chosen sample rate (8 kS/s should be high enough for human speech).