Bomberman

The goal of every stage is to defeat all enemies, then find the exit door which is hidden beneath a random soft block. Bomberman begins with only one bomb and a blast radius of one tile; one power-up item is hidden in each stage, allowing Bomberman to gain better equipment as he progresses. New enemies are introduced along the way, each with unique patterns and traits that require individual strategies to defeat.

You are given 200 seconds to complete each stage. If the timer reaches 0, all enemies on the stage will be removed and replaced with 10 Pontans.
Completing a stage will award Bomberman with an extra life. Once all lives are lost, it's Game Over.

After every five stages is a Bonus Level, wherein the goal is to destroy as many enemies as possible within the time limit. Bomberman is invincible and enemies will continuously spawn, encouraging reckless bomb usage.

If you bomb a door or an item, 8 monsters will spawn from it (and the item will be destroyed - doors can shrug off explosives). What monster is spawned depends on what item is in the stage - see the items section for each monster-item correlation.
If all the enemies in a stage can be defeated with only 7 bombs, you gain a large score bonus.

The game is 50 levels long. Completing the game will display the ending sequence before restarting at Stage 1, but with all of the power-up items you had collected by Stage 50.

Controls

Items can be found inside of soft blocks; each stage only has one item. Items are unnamed in the manual - the following names are what they're referred to in later games.

Bomb Up

Allows Bomberman to lay an extra bomb. The maximum number of bombs that can be laid is 10.
Enemy spawn: Ballom
 

Fire Up

Extends the blast radius of bombs by one tile. Maximum range is 10 tiles.
Enemy spawn: Onil
 

Speed Up

Makes Bomberman walk slightly faster. Effects are accumulative.
Enemy spawn: Dall
 

Remote Control

Bombs can now be detonated at any time by pressing the B Button; they will no longer automatially detonate.
Enemy spawn: Kondoria

Wall Pass

Allows Bomberman to walk through soft blocks. If he is standing inside a soft block while it gets hit by a bomb, it counts as damage.
Enemy spawn: Minvo

Bomb Pass

Allows Bomberman to walk through bombs.
Enemy spawn: Ovapi
 

Fire Pass

Allows Bomberman to survive his own bomb blasts.
Enemy spawn: Pass
 

Question Mark

Makes Bomberman totally invincible (to enemy collisions and bomb blasts) for 30 seconds.
Enemy spawn: Pontan

Bonus Items award points upon collection, and require an elaborate procedure to "unlock" instead of simply finding inside soft blocks. These names seem to be consistent among all online Japanese sites, though I'm not sure what source they come from - guide book, perhaps?
Credit goes to The Cutting Room Floor Wiki for instructions on how to find the bonus items.

B Panel / Bパネル

An object from Star Force (a space shooter by Tecmo, ported to the Famicom by Hudson).
Don't defeat any enemies. Then walk over the exit door.
Points: 10,000
Stages: 6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48

Goddess / ゴーデス

Another object from Star Force.
Defeat every enemy. Then walk around the entire outer ring of the level in a circle.
Points: 20,000
Stages: 1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49

Cola / コーラ

Don't defeat any enemies. Find the exit door and walk over it - keep walking without letting go of the D-Pad for approximately 15 seconds.
Points: 30,000
Stages: 4, 12, 20, 28, 36, 44

Famicom / ファミコン

The console the game is being played on. It's, like, meta or something.
Defeat every enemy, then create 248 chain reactions with bombs (ie. one bomb detonating another bomb). Using the Remote Control disqualifies you from getting this item.
Points: 500,000
Stages: 3, 11, 19, 27, 35, 43

Programmer Nakamoto / 中本プログラマー

A caricature of Shinichi Nakamoto.
Defeat every enemy, but don't destroy any soft blocks.
Points: 10,000,000
Stages: 2, 10, 18, 26, 34, 42, 50

Dezeniman / デゼニマン

The protagonist of Dezeni World (a Hudson Soft text adventure - see Hardcore Gaming 101 for details).
Destroy every soft block, but don't kill any enemies (including the enemies that spawn from the exit door). Then bomb the exit door 3 times.
Points: 20,000,000
Stages: 5, 13, 21, 29, 37, 45

Ballom / バロム

American name: Valcom
An orange balloon. Primarily moves in a slow and predictable back-and-forth pattern, but will occasionally change direction - usually when Bomberman is nearby.
Points: 100

Onil / オニール

American name: O'Neal
A blue onion-shaped creature. Faster than Balloms, Onil will move erratically and pursue Bomberman at times.
Points: 200

Dall / ダル

American name: Dahl
A red barrel that squashes and stretches. Behaves similarly to Onil, but is a little faster.
Points: 400

Minvo / ミンボー

American name: Minvo
An orange buck-toothed sphere. Faster than Dall, it moves erratically and will pursue Bomberman when they want to.
Points: 800

Ovapi / オバピー

American name: Ovape
A red cartoon ghost-like monster. It moves aimlessly and can travel through soft blocks.
Points: 1000

Kondoria / コンドリア

American name: Doria
A blue shapeless mass with a face. The slowest enemy in the game, it will pursue Bomberman and can travel through soft blocks.
Points: 2000

Parse / パース

American name: Pass
An orange sphere with vaguely animalistic features (common consensus says it's a tiger). The second-fastest enemy in the game, it mostly moves aimlessly but will pursue Bomberman on occasion.
Points: 4000

Pontan / ポンタン

American name: Pontan
An orange spinning disc with a face. The fastest enemy in the game, it moves unpredictably and can travel through soft blocks.
Points: 8000