I'm getting the following error in my nginx server error log file:

2010/04/16 13:24:16 [crit] 21974#0: *3188937 open() "/usr/local/nginx/html/50x.html" failed (24: Too many open files), client: 88.x.y.z, server: example.com, request: "GET /file/images/background.jpg HTTP/1.1", upstream: "http://10.8.4.227:81//file/images/background.jpg", host: "example.com"

...

2010/12/21 12:39:25 [crit] 20157#0: *230260 open() "/usr/local/nginx/html/50x.html" failed (24: Too many open files), client: 58.245.186.49, server: example.com, request: "GET /style/all.css HTTP/1.1", host: "example.com", referrer: "http://domain.com/x.php?..."

2010/12/21 12:39:25 [alert] 20157#0: accept() failed (24: Too many open files)

How do I fix this problem under CentOS / RHEL / Fedora Linux or UNIX like operating systems?

Linux / UNIX sets soft and hard limit for the number of file handles and open files. You can use ulimit command to view those limitations:
su - nginx
To see the hard and soft values, issue the command as follows:
ulimit -Hn
ulimit -Sn

Increase Open FD Limit at Linux OS Level

Your operating system set limits on how many files can be opened by nginx server. You can easily fix this problemby setting or increasing system open file limits under Linux. Edit file /etc/sysctl.conf, enter:
# vi /etc/sysctl.conf
Append / modify the following line:
fs.file-max = 70000
Save and close the file. Edit /etc/security/limits.conf, enter:
# vi /etc/security/limits.conf
Set soft and hard limit for all users or nginx user as follows:

nginx       soft    nofile   10000
nginx       hard    nofile  30000

Save and close the file. Finally, reload the changes with sysctl command:
# sysctl -p

nginx worker_rlimit_nofile Option (Increase Open FD Limit at Nginx Level)

Nginx also comes with worker_rlimit_nofile directive which allows to enlarge this limit if it's not enough on fly at process level. To set the value for maximum file descriptors that can be opened by nginx process. Edit nginx.conf file, enter:
# vi /usr/local/nginx/conf/nginx.conf
Append / edit as follows:

# set open fd limit to 30000
worker_rlimit_nofile 30000;

Save and close the file. Reload nginx web server, enter:
# /usr/local/nginx/sbin/nginx -t && /usr/local/nginx/sbin/nginx -s reload
# su - nginx
$ ulimit -Hn
$ ulimit -Sn

Sample outputs:

30000
10000

how to confirm how many file descriptors nginx is running with?

cat /proc/<pid>/limits