So my "essentials" list will be divided in two parts (bear in my mind that it's focused on rock, other people in this thread are more apt than me to talk about other genres).
First I'd like to talk about bands that emerged in the early 2000s (the albums will not necessarily be the ones they released back then though), I'd say that it was the start of a new era, and these bands are still seen as heavyweights on the rock scene nowadays. Most of these guys were good friends, there was a feeling of camaraderie and I think it's one thing that I really love about the rock scene in Japan.
I'll put the original names of the bands/albums in parenthesis.
ELLEGARDEN - Don't Trust Anyone But Us
Straightener - CREATURES (ストレイテナー - CREATURES)
ASIAN KUNG-FU GENERATION - Magic Disk (ASIAN KUNG-FU GENERATION - マジックディスク
ART-SCHOOL - LOVE/HATE
ACIDMAN - A Beautiful Greed
THE BACK HORN - Headphone Children (THE BACK HORN - ヘッドフォンチルドレン
ORANGE RANGE - MusiQ
Fujifabric - Fujifabric (フジファブリック - フジファブリック
BUMP OF CHICKEN - jupiter
RADWIMPS - X to ○ to Tsumi to (RADWIMPS - Xと○と罪と
BEAT CRUSADERS - P.O.A. ~POP ON ARRIVAL~
Supercar - Futurama (I guess they were already popular in the late 90s though)
Second part would be about bands that spawned later and gained in popularity between 2007 and now. I'd say that they're a good representation of what the scene looks like today.
9mm Parabellum Bullet - Termination
Sakanaction - DocumentaLy (サカナクション - DocumentaLy)
KANA-BOON - DOPPEL
Czecho No Republic - MANTLE
KEYTALK - OVERTONE
ONE OK ROCK - Zankyo Reference (ONE OK ROCK - 残響リファレンス
The Mirraz - TOP OF THE FUCK'N WORLD
androp - period
Kyuso Nekokami - Change the World (キュウソネコカミ - チェンジ ザ ワールド
Kuso Iinkai - Shu no Kigen (空想委員会 - 種の起源
the telephones - Rock Kingdom
[Alexandros] (aka
[Champagne]) - Me Do No Karate
CreepHyp - Fukikoboreru Hodo no, Ai, Ai (クリープハイプ - 吹き零れる程のI、哀、愛
the HIATUS - Keeper of The Flame
Nothing's Carved In Stone - Silver Sun
I'd say these last two are special cases since they're more like superbands. Some of their members are from bands that were popular before like ELLEGARDEN, Straightener, THEE MICHELLE GUN ELEPHANT, toe, Tokyo Jihen...