Extracting IP address from a character array returned by ESP8266 module

You can use strstr to find both quotation marks, then use strncpy to copy the correct part of the IP address into the new char array:

  char inString[] = "+CIFSR:STAIP,\"192.168.1.117\"";
  char IP[16];
  char* firstQuote;
  char* secondQuote;
  Serial.begin(115200);
  delay(100);
  Serial.println();
  Serial.println(inString);

  firstQuote = strstr(inString,"\"");
  secondQuote = strstr(firstQuote+1,"\"");
  char lengthOfIP = secondQuote - firstQuote - 1;
  strncpy(IP,firstQuote+1,lengthOfIP);
  IP[lengthOfIP] = '\0'; 

  Serial.println(IP);

but this ideally needs some checks to make sure that both quote marks actually exist and that they are a sensible space apart.
[edit]
Just realised there is a 'quicker' function, strtok() http://www.cplusplus.com/reference/cstring/strtok/
which you can use, but my example may help you understand how to do this type of thing more generally.