Where does the running configuration exist and where does the startup configuration exist?

RAM holds the running configuration file, NVRAM holds the startup configuration file.

Running configurations are stored in RAM because they are active configurations with the most up-to-date commands entered.

Start-up configgurations are stored in NVRAM. When you hit the "write memory" command, the NVRAM configurations are over-written by the running configurations thus making both exact same copies of eachother.

running configuration store in ram where as start up config in flash  rom 

from there it approach to the running config

Running configaration is saved on device memory called RAM and startup configuartion is saved on nonvolatile memory called NVRAM

Running Configuration on RAM, Startup Configuration on NVRAM

exactly as Hasan Khabbazeh answered

running configuration stores on RAM

and

startup configuration stores on NVRAM

Startup-config resides in NVRAM (Non-Volatile RAM) and running-config resides in RAM (Random Access Memory)

NVRAM retains its information even power is turned off but RAM maintains data only for as long as power is applied.