Паде́ж: case (as in: the bane of Russian students' lives)
One of the great shocks of Russian 101 — along with the first acquaintance with verbal aspect — was learning about cases (падежи). Then there was a slow, disbelieving, horrified realization of how the six cases would combine with three genders and plurals or singulars to produce what seemed like thousands of permutations, all impossible to remember.
And then you find out that there aren't six cases, there are 15 of them — or maybe even 16, depending on how you classify them.
You shouldn't worry if you slept through that lecture, especially since all this case business is highly contested. Nor should you worry about learning any additional cases. Chances are you already use them — probably even correctly. You just thought they were archaic forms or some weird exceptions to the rule.
For example, your significant other most certainly exclaims Боже! (God!) Боже is the vocative — a case used when addressing or calling someone — of Бог (God). Today you are most likely to hear the vocative case in church: господи (God, from господь); отче (Father, from отец); and Иисусе Христе (Jesus Christ, from Иисус Христос).
But some linguists assert that there is a "modern vocative case," formed by dropping the final vowel in names and relations. This is when you open the door to the dacha and shout out to your family: Мам! Пап! Оль! Маш! Алёш! (Mom, Pop, Olya, Masha, Alyosha!). However, other linguists vehemently object to this characterization. So you may be vocative, or you may just be slangy and insistent.
Another case is the locative — easily deciphered as pertaining to something's location. You know this, too. Где твой пиджак? Как ты думаешь — в шкафу! (Where is your jacket? Where do you think? In the wardrobe!) The –у is the locative ending. Compare with a different sentence and ending for the wardrobe: О новом шкафе мы пока не думаем. (We aren't thinking about a new wardrobe right now.)
At the dacha you use the locative case often: Ребята ищут грибы в лесу. (The kids are looking for mushrooms in the woods.) Пакет был зарыт в снегу недалеко от гаража. (The bag was buried in the snow near the garage.) На даче мы живём в раю. (At the dacha we are living in paradise.) Муж работает в саду. (My husband's working in the garden.)
Note that if you're talking about the woods, snow or paradise, you use the prepositional case: о лесе, о снеге, о рае, о саде (about the woods, snow, paradise, garden).
Then there is the mind-bending translative case, which you have certainly read and heard, and maybe even used. This is when someone is being transformed and acquiring a new profession, status, or elected position. It sounds a bit weird because it's the same as the plural nominative case: в президенты, в жёны, в люди (become president, a wife, an adult, i.e., a person out in the world). Он баллотируется в президенты (He is running for president).
Король сказал, что тому, кто найдёт амулет, он отдаст в жёны прекрасную принцессу. (The king said that he would give the beautiful princess in marriage to whomever finds the amulet.) Усыновила их, обучила, вывела в люди. (She adopted them, taught them, and sent them out into the world.)
Another cool case is the — no, forget it. There is one other important rule of grammar: It's only good in small doses.