Why do I need to type ./mysql even that I am in the folder? So I am in a folder where mysql executable already is..

When I try mysql in terminal I get:
-bash: mysql: command not found

But this will work:
Korays-MacBook-Pro:bin koraytugay$ ./mysql
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 107
etc etc...

So why is the ./ required? 

Also how can I add this mysql to my PATH?
The ./ is required because "." is not in your PATH. When you enter a command, the shell will search the folders listed in the PATH environment variable for that command, if you haven't given a path with the command.

If you do not list "." (the current folder) in the PATH variable, it is simply not searched.

You add it to the PATH variable as you would any other directory - it is simply a dot. 

Wheere it is set depends on which shell you use, and whether you want it set for just you or everyone. You could for example set it in the .bashrc file in your home directory or in /etc/bashrc to change it for everyone on the system.