Don’t forget to set it up as a service. This is from the tutorial @Omar posted.

sudo yum -y install mariadb-server mariadb
sudo systemctl start mariadb.service
sudo systemctl enable mariadb.service

Once you do that, you should be able to run the mysql command and it will just let you in.

[vagrant@default ~]$ mysql
Welcome to the MariaDB monitor.  Commands end with ; or \g.
Your MariaDB connection id is 2
Server version: 5.5.37-MariaDB MariaDB Server

Copyright (c) 2000, 2014, Oracle, Monty Program Ab and others.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

MariaDB [(none)]> 

So… you’ll probably want to secure it:

mysql_secure_installation

MariaDB is shipped in the CentOS repo as of CentOS 7 instead of mysql.
if you still want to install mysql you need to add mysql rpm dependency into your yum repo.

check this tutorial out of how to install mysql 5.6 in CentOS7 :

How to install latest mysql 5.6 on CentOS7

Not 100%, sure if this is true in your case but there may not be a package yet. CentOS 7 is only been out for a few days and they changed some of the inner workings in this version. If you are hosting everything on one server (web app & database), I would recommend using CentOS 6.5 now and keep an eye on 7 until package managers catch up to the new release and at that time deploy a CentOS 7 server and rsync all your files over. There also is an upgrade tool to upgrade 6.5 to 7 in the works, you also could take that action. (I would not recommend this, as there could be downtime. When the time comes I would build the CentOS 7 server and transfer everything over. This would mean 5 - 10 minutes downtime if done properly.

I’m pretty sure that mysql has been replaced by MariaDB.

Yes it has been.

Right. CentOS 7 (as well as RHEL 7) does not include MySQL. It instead ships MariaDB which a drop in replacement. You can install it with:

sudo yum install mariadb-server mariadb

Can I migrate MySQL database to Maria DB?

Is it transparent?

https://www.youtube.com/watch?v=qKdImyRaE9Q

Yes the title is wrong. This is for mariadb which is already installed.
I tried adding the yum repository as noted by rparker above ( rpm -Uvh … ) but that failed dependency checks without any detail error messages on cause

We use Centos 7 and have found Percona to meet our needs quite well whether we need a single server or a cluster.

Install Percona XtraDB Cluster on Centos 7