Erik, the string "$this->subject" refers to a variable declared and used within a class. A class is a Object Oriented Programming concept, where you group every relevent variable and function into a single object.
So imaging you write a PHP script to display a table. You have one array storing all the column headers, a second array storing all the row headers, and a third array to store the rest of the table content. You also have a procedure to use all these 3 arrays to construct a table in html.
This is not the best example, but say you have to do alot of these type of tables, you don't want to repeat the same code over and over again. What you do is store this piece of code as a class. Then you can create a instant of this class object everytime you need to output a html table.
the variable "$this" is just a simple way for the class object refer to itself.
the string "->" tells the instance you created, now you are using a variable or function in this class.
the string "subject" is just a variable you have declared in this class. For example, for any of the 3 arrays you can do "$this->array1", "$this->array2" or "$this->array3".
To access a function within the class, its very similar to accessing the class variables. You just do "$this->createHTMLtable()" assuming 'createHTMLtable' is a function you wrote to generate the html table.
Hope this helps.
-Will