How do I connect to my MySQL database server from an other server (say Apache or Tomcat app server) in same VLAN under CentOS / Fedora / RHEL / Redhat Linux?

First, you need to turn on the remote access for your database server.

Sample Setup

Consider the following sample setup:

                              +----------- server1 192.168.1.6
                              | 
                              +------------ tomcat1 192.168.1.7
                              | 
 +------------------+         |
 | MySQL Server     | --------+------------ apache2 192.168.1.8
 | 192.168.1.5:3306 | --------+
 +------------------+         |
     LAN      192.168.1.0/24  +------------- pc1 192.168.1.51
                              |
                              |
                              +-------------- pc25 192.168.1.76

You need to allow access to 192.168.1.5 from apache server located at 192.168.1.8.

Step #1: Configure MySQL Server For Remote Access

Open a terminal or login to 192.168.1.5 using the ssh command:
$ ssh root@192.168.1.5
Edit /etc/my.cnf, enter:
# vi /etc/my.cnf
Modify or append as follows:

# make sure the following line is deleted or commented out
# skip-networking
bind-address    = 192.168.1.5

Save and close the file. Restart the mysql server, enter:
# service mysqld restart

Make Sure TCP Port # 3306 is Opened For Business

Verify that the TCP port 3306 is open, enter:
# netstat -tulpn | grep :3306

Step #2: Linux Firewall Configuration For TCP Port # 3306

You need to open TCP port # 3306 at the firewall level, enter:
# iptables -A INPUT -i eth0 -s 192.168.1.8 -p tcp --destination-port 3306 -j ACCEPT
# service iptables save

Step #3: Configure Database Remote Access

You need to grant access to an existing database called salesdb from remote IP called 192.168.1.8 using a username called foo. First, connect to mysql server as root user, enter:
# mysql -u root -p mysql
Type the following command At mysql> prompt, enter:
mysql> update db set Host='192.168.1.8' where Db='salesdb';
mysql> update user set Host='192.1681.8' where user='foo';
mysql> \q
Login to 192.168.1.8 and type the following command to test mysql server remote access:
$ mysql -u foo -h 192.168.1.5 -p salesdb
Sample outputs:

Reading table information for completion of table and column names
You can turn off this feature to get a quicker startup with -A

Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 27720995
Server version: 5.0.77 Source distribution

Type 'help;' or '\h' for help. Type '\c' to clear the buffer.

mysql>