If you want to design a class for database access, I'd suggest you don't. There are already many classes that do just that. There is a PHP extension called PDO that can help you do the type of thing above and it is already well tested. Here is a link to good tutorial on it: http://www.phpro.org/tutorials/Introduction-to-PHP-PDO.html
OOP in PHP is very similar to OOP in other languages such as C++, C# (similar not the same, the basic concepts are very transferable. You should probably look into the links posted above to understand that in detail).
Edit
Let's see how this goes :). Here are the basics.
A class is a type of something. You can have a Person class, Car class etc.
An object is an instance of the class. It is one thing of that type. In a PHP context, this is how this will look:
class Person // Class
{
public $name; // Property
public function setName($n) // Method
{
$this->name = $n;
}
}
I have created a class called Person.
$p = new Person(); // An instance of Person class
I have created an instance of the class. $p is an object of type Person.
The $name inside is a member variable(/attribute). Think of it is as one of the properties that defines a Person. The Person class is a container for a set of data that defines a Person; name is one just one such data.
A class can have methods. Think of these as ways you interact with the class. You can call a method for the class to do something. In the above example, the setName method can be called with 1 argument. This argument is set as the value of the name member variable. The $this has to be used to refer to member variables (the $n is not a member variable, i.e. it is not a property of the class)
This should give you the basics to get started. Everything else builds on this.