What is the difference between "herr" and "herrn"?

Herr, in German, means mister, sir, gentleman, or in elevated cases, lord or master.

It's important to realize something about German: cases.
There are four different cases in German, and they are:
nominative - when something acts as a subject
(the person doing the action)
accusative - when something acts as a direct object
(the thing the action happens to)
dative - when something acts as an indirect object
(like accusative but preceded by to, at, in, etc.)
genitive - when something acts as possessing something.
(the person to whom something belongs)

Here's a sample sentence.
He came into the garage and stole John's bike.
The subject, he, is nominative.
The direct object, the bike, is accusative.
The indirect object, the garage, is dative.
The person possessing something, John, is in the genitive.

In English, cases don't matter very much.
However, in German they are extremely important.
German cases even change how to write and say words.
(This is called declension.)

Herr will appear as Herrn because of its declension.
In the nominative, it's Herr.
In any other case, it's Herrn. (i.e., when it's acting as the object of a sentence)
When it's plural, it's always Herren.