I'll mention one heroine, and one hero - plus some runners up for each section. Spoilers may occur.
If I had to pick one Heroine, it would definitely be Seras Victoria from Hellsing (Hellsing Ultimate OVA series in particular, or the manga it is based on).
She's powerful - a necessity for any heroine who engages in battle. She's also quite the cutie (ha), and I don't just mean her cup size. While the series in question is light on character development and on personality as a result of this, she goes through significant character growth as well.
At the start of the series, she's simply an ordinary person caught in unfortunate circumstances. Yet as she finds herself fighting monsters, and trying to deal with becoming a monster herself, you see her struggle quite thoroughly. Out of all of the characters in Hellsing, she has the most interesting development, being a clear "main character" that - while not as overbearingly "overpowered" as the male lead Alucard, is far more important to the interesting bits of the plot.
Runners up for my favorite heroine from anime go to Tsunemori Akane from Psycho Pass (as essentially a tie), with Yona from the similarly named Yona of the Dawn also being a character I quit enjoy as a heroine. I admit however that there are few female characters I've identified with in anime as much as the men (being a man myself), and this might also be as a result of the genres I enjoy mainly having more heroes than heroines. My two favorite ones however are two of my favorite characters in anime, anyway, bar none.
I would have put Emilia from Re;Zero in the running here if the anime were to run a second season and cover another season of content, but based just on what the anime has shown of her, I can't put her on the list - she just isn't around often enough.
Now moving on to heroes, this one is more difficult for me, since I have more characters that I identify with and like among these. It boils down to a choice between Natsuki Subaru from Re;Zero, and Emiya Kiritsugu from Fate/Zero. If I were allowed to include characters from manga series, I would be inclined to include Guts from Berserk in the running as well - but adaptations of that series into anime have been mediocre at best.
Of my two choices, Emiya Kiritsugu is the character who I feel more pity for, and he is also far more competent in many ways. However, he finds only all too late how his ideals are misplaced - his intention to always bring about the "greater good" leading only to the countless deaths of innocent people. This is in direct opposition to Subaru's character growth, where his flaw (and strength) is in refusing to lose anything - yet unlike Kiritsugu, Subaru is granted more leeway to make good on his desires.
Subaru is a hero that a lot of people love to hate, so my love for his character might seem odd from the perspective of many.
He begins the story as an ordinary 17 years old guy who has wasted his high school years away, and is suddenly thrust into a world where he is forced to confront atrocities and pain on a level that no human being ever has or could. Subaru falls in love, and does everything he can to prevent who he loves from dying a horrible death that only he can prevent. Indeed, this very fact - that his ability is the key to the salvation of those around him - leads to his egotism and selfish nature being brought out on full display, and these are obstacles he has to overcome which are just as important to the story as him overcoming death itself.
Subaru consistently tries to do the right thing, yet he is only human, and consistently suffers through fates worse than death (plus death itself), only barely managing to hang on to his sanity throughout. Subaru is my favorite hero character in anime, because he has the best character development of any protagonist I have ever seen. He is utterly human, as well as utterly flawed, and yet these very flaws and his own lack of ability make his every success and every failure meaningful to the extreme. He's not some stereotype: neither a genius, or prodigy, or morally perfect man. Subaru's strength as a character is in his perseverance, and in his ability to turn his selfish desires for love and for recognition into a desire to protect and save everyone that he can.
Runners up as far as hero-characters I would consider include Edward Elric from Fullmetal Alchemist, Vash from Trigun (the manga is much better), and Izumi Shinichi from Parasyte. If I had to consider other male characters who I find greatly interesting, then Light from Death Note and Makishima Shogo from Psycho Pass both are quite interesting villains - but while one of them is a protagonist, he never is in any way a hero.