Nginx listens at a port, only responds if set to port 80

Try the following server block:

server {
   listen       81 default_server;
    server_name _;    
    root    /usr/share/nginx/html;
    location / {
        index   index.html;
    }
}

The underscore _ is a wildcard, Also the *:81 likely doesn't do what you expect, just use the port number.

Then test your settings with nginx -t:

nginx: the configuration file /etc/nginx/nginx.conf syntax is ok
nginx: configuration file /etc/nginx/nginx.conf test is successful

Restart nginx:

service nginx restart

Test with netstat:

root@gitlab:~# netstat -napl | grep 80
tcp        0      0 0.0.0.0:80              0.0.0.0:*               LISTEN      7903/nginx      
tcp        0      0 127.0.0.1:8080          0.0.0.0:*               LISTEN      2662/unicorn.

Update

I installed nginx on a test system. With the stock nginx.conf file and a 1 line change to /etc/nginx/sites-enabled/default, I was able to retrieve files from port 81

cat /etc/nginx/sites-enabled/default
server {

    listen   81;
    server_name localhost;
    root /usr/share/nginx/www;
    index index.html index.htm;


    location / {
        try_files $uri $uri/ /index.html;
    }

    location /doc/ {
        alias /usr/share/doc/;
        autoindex on;
        allow 127.0.0.1;
        deny all;
    }

}

Netstat output:

netstat -napl | grep 81
tcp        0      0 0.0.0.0:81              0.0.0.0:*               LISTEN      3432/nginx

Download file:

$ wget localhost:81

Contents of file:

$ cat index.html
<html>
<head>
<title>Welcome to nginx!</title>
</head>
<body bgcolor="white" text="black">
<center><h1>Welcome to nginx!</h1></center>
</body>
</html>

Update2

Test port:

 root@gitlab:# nc -vz localhost 81
 Connection to localhost 81 port [tcp/*] succeeded!
 root@gitlab:# nc -vz localhost 443
 nc: connect to localhost port 443 (tcp) failed: Connection refused

Turns out the big problem? Nginx had set worker_processes to 0. I added a line setting it to auto in the top of my nginx.conf, and all was well with the world!

Thank you all for your time and patience.