Do you know what Object Oriented Programming (OOP) is? As a quality programmer, it's a good thing to know.
Here's a very very simple analogy of what you're looking at:
Think about a car. Now think about how many tires it has.
For simplicity sake, let's say the car has four tires. In OOP this could be depicted with variables as:
$car->numTires = 4
The command in your sql query: "mysql_fetch_object" is taking the results returned from the query, and turning them into an object. Similar to how "mysql_fetch_array", would turn the results into an array.
So "$sql->howmany" is the number of db entries in the table "mailbox" where the recipient field equals the variable $recipient.
You should invest in a beginning computer science book, and read up about OOP. The "->" terminology is used in most all C based languages (php is one of them), and the OOP methodology is a driving force behind the Java language.
If your programming career progresses, this is not the last time you will see OOP in PHP.