A Nest is a nest, and Wärme means "warmth." Put the two together and you get a word for the sense of safety and psychological comfort a family provides, especially for children (provided, of course, that the family is not dysfunctional). The term can also be used in a more general meaning to denote the comfort and sense of belonging a tightly-knit group may provide for its members.
Thanks to Wunderkind and Kindergarten, Kind (child, plural Kinder) is probably one of the best known German words to English speakers. The verb sorgen is another matter. In its reflexive form—with the preposition um (about)—it means "be concerned" or "worry" (about someone or something). As an intransitive verb—with the preposition für (for)—it means to "take care" of or "provide" for something or someone. The noun Sorge (plural Sorgen) also has the double meaning of "worry" and "taking care," and both meanings are present in our word of the month: A Sorgenkind is a "problem child" whom parents are most concerned about and who needs the most help among their children. We may say, for example, "Walter war von Anfang an ein Sorgenkind" (Walter was a Sorgenkind from the beginning). The term can also be used in a figurative sense. For example, we may say that the luxury car division is the Sorgenkind of a car manufacturer.
Note that a Sorgenkind is not the same as a schwarzes Schaf (black sheep) in the family: The former commands sympathy and receives help, while the latter most often does not.
A Bär is a bear (see my Word of the Month Seebär) and a Dienst is, in the present context, a service performed for somebody. A Bärendienst is a service that backfires—meant to benefit the recipient, it has the opposite effect; it may even turn out to be a disaster for the intended beneficiary.
Why connect a bear, among all creatures, to that type of action? There seems to be general agreement that the German term derives from La Fontaine’s fable L'Ours et l'amateur des jardins ("The Bear and the Garden Lover"). It tells the story of a lonely gardener and a lonely bear who become companions. The bear assists his friend in his work, and when the gardener takes a nap, the bear tries to ward off a bothersome fly. When all else fails, he picks up a paving stone and crushes the fly, which had settled on the gardener's nose. Alas, the blow also kills the gardener.
Nerven are nerves, and a Säge is a saw. Taken together, they refer to somebody or something that gets on your nerve, badly and persistently. A Nervensäge can be strictly a creature of the imagination, like Frosty, the Snowman, or Rudolph, the red-nosed reindeer, or something that exists in real life, like a child who's endlessly complaining or a sports commentator who is more in love with the sound of his voice than the game he is supposed to comment on.
BTW The kind of handsaw that was the inspiration for my rendering of a Nervensäge is called a Fuchsschwanz (fox tail) in German.
Morgen means "morning," and a Muffel is a sullen person, a grouch. Put together, they signify a person who habitually wakes up in a bad mood and needs some time before being able to face the world with an even temper.
The late German chancellor Willy Brandt was, according to his wife Ruth, a Morgenmuffel. And recently, I came across a study dealing with the important question why (German) teenagers are such Morgenmuffel (the plural is the same as the singular).