"It can't be helped... We'll have to use 'that'." —The Internet
The following are a list of phrases that tend to show up a lot in Japanese media. Many of them tend to be particularly common in the types of Anime and Manga works that are popular in the Anglophonic cultures and attract lots of fan translations.
Do your best! / I'll do my best! (がんばって, gambatte / がんばります, gambarimasu)
Related to the Japanese ideal of perseverance, the speaker encourages the recipient to stand strong in the face of adversity. Fortunately for translators there is a close English cultural analogue with the same number of syllables: "Go for it!"
Usually happens after an opponent's declaration that they have been underestimated which leads them to pull out all the stops and Power-Up with or without Dramatic Wind and/or Theme Music and immediately Flash Step (usually behind the opponent). Generally accompanied by an Oh, Crap face. This will many times cause the attacker to taunt with a Too Slow (遅い!, osoi!).
There's 行くぜ! (iku ze!) too, which means the same, but it's more like compelling your interlocutor to go with you (since the ze particle has imperative undertones - "you're going with me!"), while not a specific order like 行け! (ike!/ "go!"). Yuke (zo) is a slightly rougher form of the phrase, spelled with the same kanji.
Listen to what other people say! (人の話をちゃんと聞け, hito no hanashi o chanto kike)
Never give up (あきらめない, akiramenai)
No, there's still a way (手がある, te ga aru)
No way!/You're kidding! (嘘!, uso!)
Shut up! (黙れ!, damare! or うるさい!, urusai!)
Stop it! (やめて!, yamete! or やめろ!, yamero!)
Talking is useless! / Talk is cheap! / No questions asked! (問答無用, mondou muyou)
Implying, of course, that fighting is the only option.
Thank Goodness! (良かった, yokatta). Most frequently said in a relieving tone, like for example when The Hero's Love Interest sees him come back alive from his battle against the Big Bad.
You're a nuisance! / You're in the way! (邪魔だ, jama da)
You're the worst! (あなた最低です!, anata saitei desu!) Shoujo romances will typically have the girl say this to the boy.
You're (too) na´ve (甘いな, amai na)
Damn!/Shit!/Damn it! (畜生!, chikushou!; クソ!, kuso!; or the most reduced form, ちぇ!, che! - this last one is pronounced like the English interjection "tch!")
What the hell is this?/What the hell? (何だこれ?, nanda kore?) or What the hell is that? (何だそれ?, nanda sore?)
"This" or "that" doesn't have to be an object. For example, it could be used in responds to a ridiculous notion or just something ridiculous in general.
しまった (Shimatta) Many ways to translate this since a direct translation would butcher it (The Japanese language is heavily context based). The most common translation is "Shit!" and is often used when a character screws up or makes a major mistake. Could be translated as just "I screwed up" but most English speakers would simply say "Shit!" in the same situation. "Dammit", "Darn it", "Blast it", etc. more closely approximate the syllable count and may be used where a milder expletive is called for.
Just to give an example of how it works: in The Simpsons episode "Thirty Minutes Over Tokyo", after losing the family's last yen bill (turned into an origami crane) to the wind, Homer (who, like Bart, had learned Japanese in prison) says "しまった、馬鹿に!" (shimatta, baka ni!, "Damnit/Shit/Darn, how stupid!") as a caption pops up reading "D'oh!".
I can't believe it!/My word!/Holy cow!/Oh, great! (なんてことだ nante koto da!, なんてこった！ nante kotta!). It can really be translated as any interjection indicating shock.
Here is a very long list of Japanese phrases and slangs, most which are commonly used in anime/manga.
When translating anything, you can only make your translation so close to the source text before it, well, stops being an actual language you're translating into (see the quote for Woolseyism). The people who fansub anime come pretty close to this sometimes.
Sometimes, they create what Amazon.com would call "Statistically Improbable Phrases". Maybe you wouldn't be surprised if someone said them, once, in that particular situation, but if they say it every time, it's a little weird.
There are a few other aspects of Japanese that create similar translation artifacts:
いただきます (Itadakimasu): literally "I humbly receive this", it figuratively means "Thanks for the meal" and is often translated as this. Normally goes together with that "palms joined" praying gesture.
It can also be used in contexts other than dining. In Mahou Sensei Negima!, Haruna used this phrase before forcing a kiss on (unsuspecting) Negi to create a magic contract with him.
いってきます (Ittekimasu): literally "I go and come", it means "Goodbye, see you later" when one leaves home for work or school.
失礼します (Shitsurei shimasu): literally "I am being rude", it means "Excuse me" for anything from entering someone's office to walking out of a classroom
お邪魔します (Ojama shimasu): literally "I am doing an intrusion", it means "Excuse me" specifically for entering someone else's home.
お先に(失礼します)(Osaki ni (shitsurei shimasu)): literally "I am being rude [by leaving] before you", it points out the cultural imperative to stay at work or school until all the work is done.
いらっしゃいませ (Irasshaimase): literally an honorific way of saying "come", it's a greeting used by employees to customers. Typically translated as "Come in!" or "Welcome!"
お待たせしました (Omatase shimashita, or just Omatase): "Sorry to have kept you waiting", a polite formula used even when you aren't late.
待ってください (Matte kudasai): "Please wait"
ただいま (Tadaima): it means "I've arrived" (literally, "I am just here now"), used for when you enter your own house. If someone's expecting you in there, expect it to be followed by お帰りなさい (okaerinasai) (or simply okaeri - an acknowledgement of the previous quote, literally meaning "please come back in"), though some people tend to use it even if they live alone. Typically translated as "I'm home"/"Welcome home".
よろしく(お願いします) (Yoroshiku (onegai shimasu)): used in introductions or when starting to work with someone (seen frequently in manga and anime scenes where a transferred student introduces him/herself before the class). Often translated as "Nice to meet you". When used together with 初めまして (hajimemashite), which also means "Nice to meet you", it is often translated more literally as "Please treat me well".