The first thing to check is that the mysql server is running. Can you ssh into the server? Do you know what distro it uses? On some distro's you can run "sudo /sbin/chkconfig mysql". If it is off, then run "sudo /sbin/chkconfig mysql 35" for RedHat, SuSE, Mandriva, etal. If the server is running Debian or Slackware, They use different run level numbers and don't use the chkconfig command.
If you aren't sure then run: "grep default /etc/inittab"
grep default /etc/inittab
# The default runlevel is defined here
# runlevel 0 is System halt (Do not use this for initdefault!)
# runlevel 6 is System reboot (Do not use this for initdefault!)
This will tell you what the default run level is.
Make sure that mysql is running for the default run level at least. Reading the entire file, you can find which is the multiuser w/network and which is the multiuser w/network and X windows from the comments.
You can download the mysql manual from their website. Make sure you read the section on securing mysql. As you are hosting a website, you must make sure that the root account has a password. Don't assume that the hosting service has done this.
On all systems you can run "ps -A | grep mysql":
ps -A | grep mysql
2988 ? 00:00:00 mysqld_safe
3055 ? 01:06:48 mysqld
to check if the service is running.
I haven't run a debian based system since Corel Linux, which was years ago and I was a newbie at the time, so I'm afraid I don't know which run level each distro uses and whether you need to create a symbolic link in /etc/init.d/rc.X/ (where X is the run level) or if there is a more convenient tool to do this. Without knowing the distribution you are using, I would just be confusing you anyway with potentially unnecessary details.
Most distro's have a Services wizard where you can check if services are running, and enable them as well (in a GUI). This may be your first step.