How to create a connection to a SQL database in a class and use the connection on a button event in another form.

You can write class with two methods one for opening connection and second for closing. Write two methods in a class
Class Connection
{
SqlConnection cn;
public void OpenConnection()
{
cn=new SqlConnection("Your Connection String Here...");
cn.open();
}

public void CloseConnection()
{
cn.Close();
}
}

and in your form call this methods like
button_click(...)
{
//create instance of connection class
Connection clscon=new Connection();
clscon.OpenConnection();
//Do your work here.......
//
//close connection after your work finish
clscon.CloseConnection();
}