ANIME IS NOT CARTOON

Seriously, there's a difference

Yes, I know that anime is 'cartoon' is Japanese, but there's a difference, take a look at this (sure, it's blatant)

With That Aside, Here Are Some Anime That I've Watched That I would definitely recommend all views to watch.

(these aren't ranked)

No. 1 - Angel Beats

This is a synopsis : "Otonashi awakens only to learn he is dead. A  rifle-toting girl named Yuri explains that they are in the afterlife,  and Otonashi realizes the only thing he can remember about himself is  his name. Yuri tells him that she leads the Shinda Sekai Sensen  (Afterlife Battlefront) and wages war against a girl named Tenshi.  Unable to believe Yuri's claims that Tenshi is evil, Otonashi attempts  to speak with her, but the encounter doesn't go as he intended.
 

Otonashi decides to join the SSS and battle Tenshi, but he finds himself  oddly drawn to her. While trying to regain his memories and understand  Tenshi, he gradually unravels the mysteries of the afterlife."

This is a very-tear jerking anime torwards the end of the show. Even the most strongest of men will cry to this.

-http://myanimelist.net/anime/6547/Angel_Beats?q=ange

No 2 -  Ao No Exorcist 

Here's another synopsis : "Humans and demons are two sides of the same  coin, as are Assiah and Gehenna, their respective worlds. The only way  to travel between the realms is by the means of possession, like in  ghost stories. However, Satan, the ruler of Gehenna, cannot find a  suitable host to possess and therefore, remains imprisoned in his world.  In a desperate attempt to conquer Assiah, he sends his son instead,  intending for him to eventually grow into a vessel capable of possession  by the demon king.
 

Ao no Exorcist follows Rin Okumura who appears to be an ordinary,  somewhat troublesome teenager—that is until one day he is ambushed by  demons. His world turns upside down when he discovers that he is in fact  the very son of Satan and that his demon father wishes for him to  return so they can conquer Assiah together. Not wanting to join the king  of Gehenna, Rin decides to begin training to become an exorcist so that  he can fight to defend Assiah alongside his brother Yukio." 

- http://myanimelist.net/anime/9919/Ao_no_Exorcist?q=ao

No - 3 : Date A Live

Synopsis - "Before the world ends, kill me or kiss me."
 

Thirty years before the events of Date A Live, an enormous  explosion devastates east Asia and kills 150 million people. This is the  first known "Spacequake", an inexplicable natural disaster that has  since become commonplace. Fast forward to the future. High school second  year Shidou Itsuka lives alone with his cute little sister while their  parents are away.
 

What do these things have to do with each other? While rushing to save  his sister from a sudden Spacequake, Shidou is caught in the blast and,  in the midst of the chaos, finds a mysterious girl. It turns out that  this girl is actually a Spirit, a powerful being from another world  whose arrival devastates the surrounding area. Thankfully, Shidou is  rescued by an anti-Spirit strike team... led by his little sister?!
 

This vicious task force is locked and loaded, ready to exterminate  Spirits with extreme prejudice. But this violent method is not for  Shidou. He discovers the one way to neutralize these Spirits peacefully:  make them fall in love. Now, it's up to Shidou to save the world by  dating those who threaten to destroy it!"

- http://myanimelist.net/anime/15583/Date_A_Live?q=da

No 4 - Guilty Crown

Synopsis - Japan, 2039. Ten years after the outbreak  of the "Apocalypse Virus," an event solemnly regarded as "Lost  Christmas," the once proud nation has fallen under the rule of the GHQ,  an independent military force dedicated to restoring order. Funeral  Parlor, a guerilla group led by the infamous Gai Tsutsugami, act as  freedom fighters, offering the only resistance to GHQ's cruel despotism.
 

Inori Yuzuriha, a key member of Funeral Parlor, runs into the weak and  unsociable Shuu Ouma during a crucial operation, which results in him  obtaining the "Power of Kings"—an ability which allows the wielder to  draw out the manifestations of an individual's personality, or "voids."  Now an unwilling participant in the struggle against GHQ, Shuu must  learn to control his newfound power if he is to help take back Japan  once and for all.
 

Guilty Crown follows the action-packed story of a young high  school student who is dragged into a war, possessing an ability that  will help him uncover the secrets of the GHQ, Funeral Parlor, and Lost  Christmas. However, he will soon learn that the truth comes at a far  greater price than he could have ever imagined."

-http://myanimelist.net/anime/10793/Guilty_Crown?q=guil

No 5 : Infinite Stratos

Synopsis - Japan engineered an armed powered  exoskeleton "Infinite Stratos" (IS) and it became the mainstream of  weapons. Since only women can operate IS, women dominate the society  over men. Orimura Ichika is a 15-year-old boy and accidentally touches  an IS placed in the IS pilot training school. He is found to be the only  man who can operate IS and forced to enter the training school.  Ichika's busy school life surrounded by girls has begun.

-http://myanimelist.net/anime/9041/IS__Infinite_Stratos?q=i

No 6 - Mahouka Koukou no Rettousei

Synopsis - In the dawn of the 21st century, magic, long  thought to be folklore and fairy tales, has become a systematized  technology and is taught as a technical skill. In First High School, the  institution for magicians, students are segregated into two groups  based on their entrance exam scores: "Blooms," those who receive high  scores, are assigned to the First Course, while "Weeds" are reserve  students assigned to the Second Course.
 

Mahouka Koukou no Rettousei follows the siblings, Tatsuya and  Miyuki Shiba, who are enrolled in First High School. Upon taking the  exam, the prodigious Miyuki is placed in the First Course, while Tatsuya  is relegated to the Second Course. Though his practical test scores and  status as a "Weed" show him to be magically inept, he possesses  extraordinary technical knowledge, physical combat capabilities, and  unique magic techniques—making Tatsuya the irregular at a magical high  school.

-http://myanimelist.net/anime/20785/Mahouka_Koukou_no_Rettousei?q=mahouka

No 7 : Mirai Nikki

Lonely high school student, Yukiteru Amano,  spends his days writing a diary on his cellphone, while conversing with  his two seemingly imaginary friends Deus Ex Machina, who is the god of  time and space, and Murmur, the god's servant. Revealing himself to be  an actual entity, Deus grants Yukiteru a "Random Diary," which shows  highly descriptive entries based on the future and forces him into a  bloody battle royale with 11 other holders of similarly powerful future  diaries.
 

With the last person standing designated as the new god of time and  space, Yukiteru must find and kill the other 11 in order to survive. He  reluctantly teams up with his obsessive stalker Yuno Gasai (who also  possesses such a diary), and she takes it upon herself to ensure his  safety. But there's more to the girl than meets the eye, as she might  have other plans for her unrequited love...

-http://myanimelist.net/anime/10620/Mirai_Nikki_TV?q=mirai%20nikkai

No 8 - Sword Art Online

In the year 2022, virtual reality has  progressed by leaps and bounds, and a massive online role-playing game  called Sword Art Online (SAO) is launched. With the aid of "NerveGear"  technology, players can control their avatars within the game using  nothing but their own thoughts.
 

Kazuto Kirigaya, nicknamed "Kirito," is among the lucky few enthusiasts  who get their hands on the first shipment of the game. He logs in to  find himself, with ten-thousand others, in the scenic and elaborate  world of Aincrad, one full of fantastic medieval weapons and gruesome  monsters. However, in a cruel turn of events, the players soon realize  they cannot log out; the game's creator has trapped them in his new  world until they complete all one hundred levels of the game.
 

In order to escape Aincrad, Kirito will now have to interact and  cooperate with his fellow players. Some are allies, while others are  foes, like Asuna Yuuki, who commands the leading group attempting to  escape from the ruthless game. To make matters worse, Sword Art Online  is not all fun and games: if they die in Aincrad, they die in real life.  Kirito must adapt to his new reality, fight for his survival, and  hopefully break free from his virtual hell.

-http://myanimelist.net/anime/11757/Sword_Art_Online?q=sword

No 9 : No Game No Life

The mysterious god welcomes Sora and Shiro to Disboard, a world where  all forms of conflict—from petty squabbles to the fate of whole  countries—are settled not through war, but by way of high-stake games.  This system works thanks to a fundamental rule wherein each party must  wager something they deem to be of equal value to the other party's  wager. In this strange land where the very idea of humanity is reduced  to child’s play, the indifferent genius gamer duo of Sora and Shiro have  finally found a real reason to keep playing games: to unite the sixteen  races of Disboard, defeat Tet, and become the gods of this new,  gaming-is-everything world.

-http://myanimelist.net/anime/19815/No_Game_No_Life?q=no%20game

Honorable Mentions

-Trinity Seven

-Oregairu

-The World God Only Knows

-Highschool Dxd

-Shinmai Maou no Testament 

-Noucome

-Nagi No Asukara

-Log Horizon

-Fate/stay Night

-Danmachi

-Accel World