Heart Eyes

In anime, sometimes a character's eyes change into hearts. The term for this in Japanese would be haato me ハート目, "heart eyes," but other phrases can be used to describe it, like me ga haato 目がハート, "the eyes are hearts," or me ni haato 目にハート, "hearts in eyes."

This includes heart-shaped pupils and irises, too. Some related words are haato maaku ハートマーク, "heart mark," i.e. ❤, and haato-gata ハート型, "heart-shaped."

Souryuuin Akemi 奏流院朱美, example of heart-shaped pupils.

Character: Souryuuin Akemi 奏流院朱美
Anime: Dumbbell Nan Kilo Moteru? ダンベル何キロ持てる？ (Episode 1)

Usage

Love

As one would expect, this sort of eye normally shows up when a character is in love with another character.

Character: Kukuri ククリ
Anime: Mahoujin Guruguru 魔法陣グルグル (2017) (Episode 5)

Background: suki スキ (好き), "[I] love [him]," a bunch of times.

Tsuchida Kanae 土田香苗, example of "heart eyes," haato-me ハート目.

Character: Tsuchida Kanae 土田香苗
Anime: Amagi Brilliant Park, 甘城ブリリアントパーク (Episode 8)

Selen Hemein セレン・ヘムアエン, example of heart-shaped pupils.

Character: Selen Hemein セレン・ヘムアエン
Anime: Tatoeba Last Dungeon Mae no Mura no Shounen ga Joban no Machi de Kurasu You na Monogatari, たとえばラストダンジョン前の村の少年が序盤の街で暮らすような物語 (Episode 1)

Katarina Claes カタリナ・クラエス, example of heart shape at the bottom of the iris.

Character: Katarina Claes カタリナ・クラエス
Anime: Otome Game no Hametsu Flag Shika Nai Akuyaku Reijou ni Tensei Shiteshimatta…, 乙女ゲームの破滅フラグしかない悪役令嬢に転生してしまった… (Episode 8)

An unusual example of highlights at the bottom of the eye drawn as hearts, rather than the pupils at the center.
Other examples from the same anime:

Love At First Sight

This also happens with love at first sight, which is pretty much how heart eyes are normally used in western animation.

Tsukino Usagi 月野うさぎ, example of heart eyes.

Character: Tsukino Usagi 月野うさぎ
Anime: Bishoujo Senshi Sailor Moon, 美少女戦士セーラームーン (Episode 1)

hito-me-bore
一目惚れ
Love at first sight.
Single-eye falling-in-love. (literally.)
- horeru
  惚れる
  To fall in love with. (becomes boreru due to rendaku 連濁.)

Katsuragi Kamui 葛木カムイ, example of heart eyes.

Character: Katsuragi Kamui 葛木カムイ
Anime: Cardfight!! Vanguard (Episode 5)

Shibata Genzou 柴田源蔵, example of heart eye.

Left: Shigure シグレ
Right: Shibata Genzou 柴田源蔵
Anime: Hataage! Kemono Michi 旗揚!！けものみち (Episode 2)

Lust

The heart eyes aren't used only with pure love. It's also used with impure love, when perverted characters be lewding on other characters. This, too, happens sometimes in western animation.

Character: Ristarte リスタルテ
Anime: Shinchou Yuusha 慎重勇者 (Episode 11)

Nosebleed and drool are symbols of lustful thoughts.

Uzume うずめ (female), versus Veronica Vasilyevna Vonitsky ヴェロニカ・ワシーリエヴナ・ヴォイニーツカヤ (male), in a reference to the reaction guys meme, known in Japan as gaijin yonkoma 外人四コマ.

Top: Uzume うずめ (female)
Bottom: Veronica Vasilyevna Vonitsky ヴェロニカ・ワシーリエヴナ・ヴォイニーツカヤ (male)
Anime: Onigiri 鬼斬 (Episode 4)

So far, the way heart eyes are used in Japanese anime is pretty much identical to how it's used in western animation. There is, however, one way which heart eyes are used in Japan that isn't shared with the west.

Extrapolating from lewding, heart-shaped pupils are sometimes used to indicated lust and are generally associated with sex. They're often combined with ahegao アヘ顔 in pornographic (hentai) illustrations, doujinshi 同人誌, etc.

Naturally, these aren't really common in your average anime because scenes depicting sex aren't really common in your average anime, but they do show up from time to time.

Yame Yukana 八女ゆかな, example of heart-shaped pupils.

Character: Yame Yukana 八女ゆかな
Anime: Hajimete no Gyaru はじめてのギャル (Episode 1)

Context: the main character imagines himself getting laid with a girl.

Single Heart Eye

In older manga and anime, sometimes eye symbols were applied to a single eye, rather than both. For reference, an example of a character with just a heart shape in just one eye:

Manga: Glass Mask, Garasu no Gamen ガラスの仮面 (Chapter 10, 冬の星座)

Context: the audience watching the theater play Futari no Joou ふたりの女王, "the two queens," is charmed by the adorable queen Ardis.
kirei dashi, yasashii shi, akarui shi, kawaii shi...
きれいだし　やさしいし　明るいし　可愛いし･･･
[She] is pretty, is gentle, is cheerful, is cute...
- akarui
  明るい
  Bright. Well-lit. The opposite of dark.
  Cheerful.
ore, mune-kyun! daze
オレ胸キュン！だぜ
[I'm in love!]

Permanent

It's possible that there's some character out there with permanent heart eyes as part of their design in anime. I haven't found one yet, though, but I did find one in a game by a Polish developer.

Character: Modeus, the Lustful demon
Game: Helltaker, ヘルテイカー

Context: Modeus' permanent heart eyes are part of her lustful demon design.

Non-Japanese

There are examples of heart pupils being used similarly in anime based on non-Japanese sources. I haven't checked if the manga which they're based on also have heart pupils, or they were added only in the anime production. For the sake of reference, I'm including them here.