Deutsche Sprache, schwere Sprache is a saying in broken German ("German language, hard language") used ironically when difficulties with German come up in a conversation.
So, here's what makes German a difficult language:
1.Gender
In German, each noun has a gender. A word can be masculine, feminine, or neuter. That means that instead of just using "the" for every word, you have three different possibilities if you want to say "The ______ is big." Masculine words use der, feminine words use die (pronounced dee), and neuter words use das.
2. Cases
There are four different cases in German, which indicate the role a word plays in the sentence. You can usually tell which case a word is in based on the article(a word like "the" or "a") in front of the word. That means that there are four ways to say "the" in front of a masculine word, 3 in front of a feminine word, 3 in front of a neutral word, and 3 in front of a word in the plural form.
That means, for the masculine word "man", you would say "the" four different ways, depending on the role of the word in the sentence.
Der Mann ist groß.
The man (subject) is tall.
Ich mag den Mann.
I like the man (direct object).
Ich gebe dem Mann eine Blume.
I give the man (indirect object) a flower.
Die Frau des Mannes ist nett.
The wife of the man ("of" relationship) is nice.
Changing the article (the word "the" in this case) according to the case is called declining a word. In Mark Twain's famous book "The Awful German Language", he says "I would rather decline two German beers than one German noun."
The gender and case working together result in a sometimes very difficult process of trying to decide how to say a word as simple as "the".

Sometimes I'm just a little confused.
3. Adjective endings
Depending on the aformentioned gender and case of a word, along with the article, you have to change the way you end the adjective. Examples using the German word gut, which means "good":
Der gute Mann or den guten Mann
Ein guter Mann or einen guten Mann
4. Plurals
The German words in plural form are much more irregular than the English. You can't just stick an 's' onto the end of a word to make it plural. There's somewhere around 12 different categories of plural words, plus exceptions.
Der Zug and die Züge (the train and the trains)
Die Zeitschrift and die Zeitschriften (the magazine and the magazines)
Der Spieler and die Spieler ( the player and the players)
All of that adds up to many small mistakes for those learning German, but they can get by anyway. A mistake in the adjective ending or using the wrong gender doesn't mean people don't understand what you say, it just means you make mistakes and you always talk a little funny.
Another effect of the complexity of the language is, in my opinion, a larger barrier between classes in German society. Often the people with a lower level of education will simply leave out the word "the" for whatever reason. Maybe it's because they don't feel like declining it, maybe it's because that's the way they always hear it in their group of friends. In any case, it results in very interesting sentences like "Come train station" or "I drive car to airport", which, quite frankly, sound a little bit neanderthal.
Does anyone else notice effects of the complexity of the German language on German society?
For all of you people out there learning German, what do you find is the hardest part about the language?