SAP Classes are much the same as classes within any other development language and offer you an object orientated way of developing your SAP application. If you are new to OO concepts then there are many resources available where you can try an learn about them.
But on a very simplistic how do I get a class/method to work in my code they are the similar to SAP function modules. The class is like the function group and the method is the actual function module that is called in order to perform the actual processing.
The ABAP code is also quite similar, only real difference is that you also have to reference the class as well as the method name.
---Function Module: CALL FUNCTION 'GUI_DOWNLOAD' EXPORTING... ---Class/Method: CALL METHOD CL_GUI_FRONTEND_SERVICES=>GUI_DOWNLOAD EXPORTING...
You will also notice like with the example GUI_DOWNLOAD that although SAP have converted many of there function modules
into class methods all they do within the method is call the original function module.
Popular SAP Classes
Here are a few links to popular SAP class objects but easiest way to find the class you are looking for is to use the search functionality on this page.
Other SAP classes
Index list to available SAP Classes