Huhn vs. Hahn vs. Hähnchen
The German words Huhn, Hahn, and Hähnchen all relate to the word chicken, but they are not interchangeable. Their usage depends on the animal's gender, age, and whether one is referring to the live animal or the food.
Huhn
A1Huhn is the generic term for a chicken. It is often used to refer specifically to a female chicken (a hen). It can also refer to the species in general or to chicken meat, especially for soups (as a Suppenhuhn). Its plural is Hühner.
Das Huhn legt jeden Tag ein Ei.
(The chicken lays an egg every day.)
Auf dem Bauernhof gibt es viele Hühner.
(There are many chickens on the farm.)
Meine Großmutter macht die beste Suppe aus einem Huhn.
(My grandmother makes the best soup from a chicken.)
Ein Huhn scharrt im Dreck nach Würmern.
(A chicken is scratching in the dirt for worms.)
Hahn
A2Hahn specifically and exclusively means a male chicken: a rooster or a cockerel. It is almost always used for the live animal, not for food. Its plural is Hähne. The word can also mean tap or faucet, but that is a different context.
Der Hahn kräht laut bei Sonnenaufgang.
(The rooster crows loudly at sunrise.)
Der stolze Hahn hat ein prächtiges, buntes Gefieder.
(The proud rooster has magnificent, colorful plumage.)
Man kann den Hahn an seinem roten Kamm erkennen.
(You can recognize the rooster by its red comb.)
Auf dem Kirchturm sitzt ein goldener Wetterhahn.
(A golden weathercock sits on the church steeple.)
Hähnchen
A1Hähnchen is the diminutive form of Hahn (literally little rooster). However, its primary and most common meaning by far is chicken prepared as food, especially a roasted or fried chicken (broiler). This is the word you will see most often on a restaurant menu. The plural is the same as the singular: Hähnchen.
Ich bestelle ein halbes Hähnchen mit Pommes Frites.
(I'll order half a roast chicken with french fries.)
Im Supermarkt gibt es frisches Hähnchen für das Abendessen.
(There is fresh chicken at the supermarket for dinner.)
Wie lange muss das Hähnchen im Ofen bleiben?
(How long does the chicken have to stay in the oven?)
Am Imbisswagen riecht es lecker nach gegrilltem Hähnchen.
(It smells delicious like grilled chicken at the food truck.)
Summary
In essence: use Huhn for the live animal, especially a hen. Use Hahn exclusively for a live male rooster. Use Hähnchen when you are talking about chicken as food that you eat, particularly roast chicken.