Bomberman

The goal of every stage is to defeat all the enemies, and then find the exit warp which is hidden beneath a random soft block. Bomberman begins with only one bomb and a one-tile blast radius.
This instalment pretty much duplicates the formula of the original NES game; a lot of what you see here will be regurgitated information. However, it does introduce new enemies, boss battles, and stages with vertical scrolling.

Each stage contains one power-up item and a set number of enemies. If the item has not yet been found once all enemies have been destroyed, the soft block containing it will flash. The block hiding the exit warp has no such luxury, however.
If you bomb a door or an item, 8 enemies will spawn from it (and the item will be destroyed - doors can shrug off explosives). Only an enemy from the current round will be spawned; see the stages page for details.

You are given 4 minutes to complete each stage. If the timer reaches 0, all enemies on the stage will be removed and replaced with 12 Pontans.

There are eight rounds with eight stages in each, making for 64 levels; the eighth stage of each round is dedicated to a boss battle, a fight against a foe with advanced tactics and multiple hit-points.

Battle

Details on the Battle Mode can be found here.

Controls

Item names are sourced from the Japanese manual. Any collected Bomb Ups and Fire Ups will always stay active, even after death or continuing from a Game Over. All items are worth 1000 points.

Bomb Up / 爆弾数アップ

Allows Bomberman to lay one extra bomb. Maximum of 10.

Fire Up / 火カアップ

Extends bomb blast radius by one tile. Maxium of 15.

Speed Shoes / 快速シューズ

Increases walking speed slightly.

Detonation Switch / 爆破スイッチ

Bombs can now be detonated at any time.

Bomb Pass / 爆弾通過

Allows Bomberman to walk through bombs.

Wall Pass / 壁通過

Allows Bomberman to walk through soft blocks.

Fire Suit / ファイアースーツ

Renders Bomberman invincible to bomb blasts for 60 seconds.

1UP

Grants the player an extra life.

Skull Item / ドクロアイテム

Exclusive to Battle mode. Curses the player with a random ailment. Curses can be spread by touching other players. The Skull will expire after 30 seconds.

• Fast movement (movement speed is tripled)
• Slow movement (movement speed is halved)*
• Weak bombs (blast radius is reduced to 1 tile)
• Bomb diarrhoea (bombs are laid at every step)
• Bomb constipation (cannot lay bombs)

Bonus Items award points upon collection, and require an elaborate procedure to "unlock" instead of simply finding inside soft blocks; they appear on a random part of the map.
I haven't seen an official source that mentions these, so their legit names are unknown. Kudos to Tadaima Bomberman Land and Arcade Gear for instructions on how to find these!

PC-Engine SuperGrafx

The upgrade to the PC-Engine that floundered at the market. Now it adds to your score!
Defeat all enemies in the stage without destroying any soft blocks.
Points: 39800

PC-Engine Core Grafx

A later model of the PC-Engine with different video input.
When the stage begins, don't move for 3 minutes.
Points: 24800

PC-Engine Shuttle

Another console variant.
Destroy all soft blocks without harming any enemies.
Points: 18800

Dr. Mitsumori

The creator of Bomberman. At least, he is until the next retcon. Rocks a fine beard.
The soft block hiding the exit must be destroyed last. I hope you like save states!
Points: 111110

Lisa

The young damsel in distress. What's her personality like? Nobody knows.
Leave only 1 enemy alive. Then stand on the exit and wait until the timer reads 3:33.
Points: 123450

Saxophone

A single-reed wind instrument. You can't play Baker Street without it.
Get to the end of the game without losing any lives.
(yes, i've yet to verify this one for myself - why do you ask?)
Points: 10000000

This game is the first in the series to the include the famous five-player battle mode! Your goal is to defeat all other players and be the last Bomberman standing! All players begin with one bomb and a two-tile blast radius.
Battle can be played with up to five players. Victory can be determined by one to five victories. Alongside the Normal Mode, there is Skull Mode, where Skull items can be found inside soft blocks (see the Skull item listing for details).

There are a few quirks, though; namely, there are no options. Or, more exactly, the options are determined by how many players are participating. See this table:

Yes, curiously, there are enemies in the arena. They will follow their usual patterns (see the Stages page for details) and can kill players on contact. They can be killed or completely ignored - they serve only as an additional threat.

There are no computer-controlled players - the Battle can only be played with human input.
The only items to be found are Bomb Ups and Fire Ups (and Skulls if Skull Mode is activated).
There is no Sudden Death. Play remains normal until the timer has fully expired, in which case all surviving players are killed and the match is considered a Draw.

Level passwords

Passwords are gleaned from the Bomberman Maniax guide book. These will work on all versions of the game barring the Amiga version.

Special Select Room

This requires a few steps to access, so follow closely!
(1) Begin the game, and get a high score of 300.
(2) Hold Run and press Select to reset the game.
(3) Wait on the title screen until the demo appears. Wait for the demo to finish. Don't press any buttons.
(4) Anytime before the menu appears on the title screen, hold Select, down, I and II. Keep holding them!
(5) When the selections appear, press Run to begin the game.
(6) On the intro cinematic, hold the Run button.
(7) If the screen above appears, you've done it right! You can let go of the buttons now. If the map screen appears, you've done it wrong: press Run & Select and try again from step 3.

This screen (referred to as a debug mode on most Japanese sites) will let you select any world, any stage, with a custom asssortment of power-ups. Press Button I to increase, Button II to decrease, and RUN Button to start.

Sound Test

On the password screen, enter "BOMSOUND". It will register as incorrect. Enter any other valid password, and you will be taken to a sound test.
Buttons I and II move back and forward through the selection, and the Run Button will play the sound or music track.