No Dreams, Only Tears
While crying is a very normal human emotion, the anime version is extremely exaggerated. There are a variety of tears in anime, which run the gamut from manly single tears to geysers that would be at home in Yellowstone. Either way, tears in anime are often drawn with extremely high surface tension, so they stick together in a tangible mass, rather than simply spread out on your face. There's also a lot more of them; a real person would probably lose 50% of the water in their body crying like Elsie (Kami Nomi zo Shiro Sekai) does.
Some tears, like One Piece's Luffy's, are accompanied by extremely runny noses. It's funny because the snot doesn't actually seem to fall anywhere; they just hang around until the character sniffs them back up or blows them noisily into a tissue.
Sailor Moon, aka Usagi, rivals her friend Sailor Mercury in water-power.
So far we've seen tears used in anime for comedy. These characters are pouting, throwing tantrums, or perhaps crying because they're touched. The genuinely pathetic tears are illustrated differently. These are clear, shiny, and oh-so sad. Facial expression is key in crying intended to illicit empathy from viewers.
AnoHana's Naruko (Anaru) here tries to fight back her tears. You can see her grimace and shut her eyes tight. Her out-of-character sobbing pulls on viewer's heartstrings. (Not that her crying is the only thing in AnoHana to break viewer's hearts.) While her tears are copious indeed, it's important to note that because they are not "silly" tears like before, they're crystalline and flow down her face, dripping from the chin like tears in real life. Not that this isn't still a very anime emotion.
Emotional expression in anime is also about body language. Suguha (Sword Art Online) curls up into a little sad ball, silently crying to herself. Her sense of loneliness and emptiness is palpable. Plus, I'm sure we've all been here at some point, perhaps even after finishing an anime binge. Here, the amount of tears is not the important part, it's the overall aura of sadness she projects.
Eyebrow Twitches and Helicopter Arms
My high school chemistry teacher had a big temper, and was always ticked off at something. You could tell when he was about to blow, because his neck would twitch violently. But that's about as anime as real life gets－eyebrow twitches are uniquely an anime emotion. In fact, real-life eyebrow/eye twitches are sometimes (rarely) an indication of serious diseases like Parkinson's. For Shugo Chara's Amu, however, it's just an indicator of her thinly veiled anger.
Other displays of anger and annoyance include grimaces, extremely slanted eyebrows, and the famous (also highly unrealistic) popping veins that are an instant indicator of emotion even when you can't see the character's face. Anime emotions really like to drive it home; there's no subtlety to be found here. Which makes sense, because unlike real humans who can exhibit a wide scale of micro-emotions, 2D folk are more limited by their simple lines. Not to mention it's not nearly as entertaining.
Aside from the standard ">:[" facial expression, Sasuke Tsubaki from Sket Dance also has steam rising up from his face. Because it's normal for you to get so hot that you generate steam in non-subzero temperatures, at least in anime.
Another thing worth noting is the shape of the eyes. In expressing anime emotions, eyes are vital, given that they often take up a quarter or more of the character's face. When people are angry or annoyed in real life, they often narrow their eyes, but in order to exaggerate this effect, anime characters are drawn with much smaller irises as well, like Sakura from Cardcaptor Sakura. Sometimes, the iris even disappears.
Some more energetic anime characters also express their anger with their arms and legs. Edward Elric from Fullmetal Alchemist is fed up with getting mixed up with his brother Al. His arms and legs move so fast they become a blur, but if you really look at the directions his appendages are moving (some sort of angry jumping-jack? Angry snow angel?), it's clear that this expression of emotion in anime would not only look super stupid, but is actually impossible to perform while standing.
Ed has no shortage of different ways to express anger. Characters will often shake their fists or grasp at the air to show their frustration.
There's also the hyper-adorable puffing of the cheeks to express mild annoyance. This isn't necessarily an anime-only expression, but it's engineered to be endearing even while showing your displeasure, which is perfect for cute anime girls like Ai Fuyuumi (OreShura). Ai exhibits a side-pout, where only one of her cheeks puff out.
Gosick's Victorique's double-sided pout is aww-inducing.
Lastly, we have terrifying walls of fire that spawn behind the character, in this case Biscuit from Hunter x Hunter. If only this was plausible in real life. Imagine how easy it would be to cook lunch!
ACTION LINES FOR MORE ACTION
Confusion and shock are also comically exaggerated in anime. Watch Ritsu from Sekaiichi Hatsukoi shake back and forth frantically, blue lines over the upper half of his face. The lines are meant to indicate that blood has left his face, leaving it pale/blue-tinged. Blood is what gives happy faces their natural flush, after all! If someone actually looked rapidly left and right like that, they'd probably get really dizzy. But in anime, it's a great way of emoting how directionless Ritsu feels.
The level of surprise can also be increased using backgrounds. The wooshing "speed lines" surrounding the characters from Nichijou and Danshi Koukousei no Nichijou make the scenes dynamic, and imply strong emotion whereas their faces indicate wide-eyed surprise.
As when expressing other anime emotions, facial parts are exaggerated. They take the term "jaw dropping" seriously in anime－Yuru Yuri's oft-forgotten main character Akari has a wide range of shocked faces, and her mouth is so big in all of them that they take up half her face.
Sawako from Kimi ni Todoke exhibits another common anime emotion trope, the sideways lightning bolt in case of epiphanies.
Do you have a fever or...?
Key to anime emotions are the blushes/flushes that cover characters faces when they are embarrassed, happy, or sometimes just to show that they're just as kawaii uguu as Nisekoi's Kosaki.
While flushing pink/red is a perfectly normal thing, especially for paler folk, the perfect oval blushes or diagonal blush marks are present only in the 2D world.
Sometimes their entire face glows so red they turn iridescent. Touka from Date A Live isn't sick, she's just really happy. Note the white bubbles that appear around her.
Again, anime emotions use body language to really drive the point home. Beloved (or hated) tsundere Taiga from Toradora! is so embarrassed she's quaking. Note that only her face is tomato-hued; her arms and knees are every bit as pale as they usually are. (And for the record, I'm totally Team Taiga.)
Gou from Free! doesn't change colors, but her hint of redness and timid hiding of her face are just as endearing.
Characters who don't show their real emotions often are portrayed not facing the camera. Only Houtaru's (Hyouka) profile is visible as he blushes and smiles to himself.
Another way of expressing shocked embarrassment in anime is having a wavering mouth. Just try to do what Aya (Kiniro Mosaic) or Hinata (Haikyuu!) do when caught in the act. Bonus: Hinata's hair also glows even redder when he's flustered.
While the origins of many anime emotions are obvious, others are so specific or divergent from reality that it makes you wonder what the original artists were thinking. But the way these anime characters express emotion is a major part of why some people like anime. After all, the graphics of an anime are such an important part of storytelling!