This is a fantastic expression that we really don’t have a match for in English. It’s said to thank someone for their hard work after the completion of a task or as a farewell to one’s colleagues at the end of the work day. My JET friend Josh once said it was best translated as, “You’re a Rockstar!” and I think that captures the sentiment perfectly.
“Huh?” “What?” “No way?” “Really?” “Are you *%$# kidding me?”
It is very unlikely you will go a day in Japan without hearing this phrase. It’s used phatically (socially) to express interest, surprise, disbelief, or in the case of Japanese students, to express dismay upon being given an assignment. “Eeeee all you want kiddos, you have to practice for the speech contest just like everyone else.”
“Please”
Learn “please” in as many languages as you can. Japanese is so cool; it’s got one for formal (onegaishimasu) and informal situations (kudasai).
“Recommendation”
Looking for something transcendent but having trouble deciphering a menu? “Osussume wa nan desu ka?” nets you the staff’s recommendation. It’s rarely failed me. Japan’s got you covered, friends.
“It’s up to you.”
Another way to ask for a recommendation, omakase shimasu is to “entrust” one’s meal to a chef in a restaurant or to defer to someone else when making a decision.
“Omakase” also retains its first usage in some American sushi restaurants.
“What?”
“Nani” is damn useful. It can be used purely for information seeking, as in nani kore (“What’s that?”) or socially with friends/family, NaaaNNIII??!!??!! (What the hell is your problem?)
“Difficult”
Aside from hearing that English is muzukashii or worse, muri (impossible) you’ll often hear this phrase used in place of “no” (usually preceded with air being sucked through the teeth.) It’s often said that the Japanese don’t like to give a direct “no,” so to say something is chotto muzukashii “a little difficult” allows you to save face by recognizing your request is unlikely to be fulfilled, while the speaker saves face by avoiding a direct refusal. Yay face-saving!
“Poor thing.” “Too bad.” “It’s a pity.”
A homophone of both cute (kawaii) and scary (kowai) and often a cause of confusion for new arrivals, this is a phrase for commenting on unfortunate incidents. It can be applied to anything from a slight mishap to a full-scale tragedy.
“Nothing can be done (about it).” “It can’t be helped.”
Shoganai can be a good or a bad thing. Similar to the serenity prayer, it encourages one to see things for what they are and accept what can’t be changed. The downside is that it can become a self-fulfilling prophecy or a justification for inaction. Along with the similar expression, shikata ga nai (literally, “There is no way to do it”) it can also be used to dismiss a difficult or unpopular course of action as “completely impossible.”
A succinct, yet thoughtful write-up on shoganai’s use in Japanese culture can be found here.
“Please be kind to me.” “I am in your debt.”
Another lovely construction that lacks a corresponding sentiment in English, this phrase is used in introductions to promote a good relationship, and when requesting a favor of someone.
“Do your best.”
Your best friend has a test coming up? Sister has to make her closing remarks at the trial tomorrow? You and dad are cheering on the Softbank Hawks? Ganbatte can be used in each situation, and stresses personal effort rather than luck. It’s also often frequently translated into English as “Fight!” but I try to steer my students away from oversimplifying it to a sentiment reserved for sporting events.
Want to politely offer encouragement to your boss or higher-up? Ganbatte kudasai formalizes it up a bit.
“Thank you.”
A very cool linguistics professor recently drew my attention to the literal translation of “arigatou” as “difficult to be.” It effectively implies that the act one is thanking for could not have been done without the person(s) being thanked. It’s poetic, elegant, and beautifully Japanese.
Doumo is a wonderful word that can be used in a variety of contexts. It can be an informal greeting to friends or amiable colleagues, while also doubling as an informal “thanks.” It can even be used to say goodbye, especially when expressing thankfulness for a visit, gift, chat, favor, etc…
“Delicious,” “Good”
Whole dinner conversations can be had using solely the two words above. Don’t forget to precede them with a contented, albeit slightly tortured sigh indicating that you are eating something SO GOOD IT HURTS. Pro tip: umai is a bit informal, so when in doubt, use oishii.
“Not that dehydrated MSG brick you wasted precious time and money eating in college.”
Learn this word and unlock a key to the very essence of being. You’ll thank me once the bliss has faded.
“Allergy”
Dietary preferences and restrictions aren’t as catered to in Japan as they are in the West. “You can’t eat meat? How about chicken?”
Fortunately, when sampling local fare, sneakily combining this word with whichever food item you find undesirable can usually help you avoid it (i.e. Gyuniku no arerurgi ga arimasu—I’m allergic to beef).
“Sweet, sweet time off.”
If you ever plan to live in Japan, you MUST know the difference between paid leave (nenkyuu) and sick leave (byokyuu). Along with unpaid overtime and rarely used vacation days is often an unspoken expectation that one will use those days when calling in sick. Few things are worse than languishing fetal on your futon, feebly insisting you don’t want to sacrifice a precious vacation day when you’ve got plenty of perfectly good sick leave accumulated for your most recent bout of infuru (flu). 
h/t: @jetset_amelia. Check out her list of must-know Japanese words here.