Listing the most popular of anime wouldn't be the wisest choice for recommendations since most people have already seen it. I always am looking for something new, so try and bring something fresh to your recommendations.
My Top 5
-26 years ago, in a third-year classroom of a middle school, there was a student named Misaki. As an honors student who was also good at sports, she was popular with her classmates. When she suddenly died, her classmates decided to carry on as if she was still alive until graduation. Then, in the spring of 1998, a boy named Sakakibara Kouichi transfers to the class, and he grows suspicious of the fearful atmosphere in that classroom. In particular, there is a beautiful, aloof girl named Mei Misaki who wears an eyepatch and is always alone drawing pictures.
2.) Guin Saga
-The ancient kingdom of Parros has been invaded by the armies of Mongaul, and its king and queen have been slain. But the "twin pearls of Parros," the princess Rinda and the prince Remus, escape using a strange device hidden in the palace. Lost in Roodwood, they are rescued from Mongaul soldiers by a strange leopard-headed man, who has no memories except for the words "Aurra" and "Guin," which he believes to be his name.
3.) Jyu Oh Sei
-Thor and his twin brother, Rai, are cast off to a forsaken planet by unknown masked men after witnessing the suspicious death of their parents. Given only a beam knife and a smelly satchel, they wander together in search of civilization on the primarily plant based world. They soon discover the plants have the advantage. Only the strong survive; the ones who throw away their humanity and become the beasts humans once were.
-A school love comedy. Kotoura Haruka is a 15-year-old girl who can read people's minds. She has been suffering from troubles caused by her mind-reading ability, and her parents got divorced as a result. She moves to a new high school but tries to keep away from her classmates. Manabe Yoshihisa, one of her classmates, accepts and appreciates her ability and she begins to interact with her friends with his help.
5.) Gunslinger Girl
-In the heart of Italy, the Social Welfare Agency rescues young girls from hospital beds and gives them a second chance at life using the latest in cybernetic advancements. With their artificially enhanced bodies, the girls are brainwashed and trained as assassins to carry out the dirty work of the Italian Government. Despite all the modifications, they are still just children at heart, struggling for recognition from those they love, even knowing the love they feel is manufactured. This tragic tale unfolds as these girls grapple with their emotions in an agency that treats them as nothing but ruthless killers.
I'll probably post more recommendations later but these are my top 5 for now. Post some of your own recommendations as well.