There is a chance for running programs on port 8081. If you install McAfee antivirus then it's agent will be running on port 8081. So, We can't use the same for other programs.
First we have to verify that the port 8081 is listening or not.
In Windows,
Open CMD and run the command to check whether any program running on the port 8081.
netstat -ano | findstr 8081
Assume that if you got output something like this,
TCP 0.0.0.0:8081 0.0.0.0:0 LISTENING 5800
Then kill the program with ID 5800 using the following command,
taskkill /pid 5800
In Linux / Mac,
Open terminal and run the command to check whether any program running on the port 8081.
sudo lsof -i :8081
Assume that if you got output something like this,
COMMAND PID USER FD TYPE DEVICE SIZE/OFF NODE NAME
node 59 root 18u IPv4 609033 0t0 TCP localhost:8081 (LISTEN)
Then kill the program with ID 609033 using the following command,
sudo kill -9 59
Alternatively,
You can also run the packager on another port.
react-native start --port=8088
react-native run-android --port=8088