
Apple Store Link US$0.99 for the Plus/HD Version
Main website: http://www.janryumon.com/
The mobile version is also available for the Android though I don't know how it compares.
Basic info on the game and a video of it playing: http://www.janryumon.com/info/index
Screenshots of the various features: http://www.janryumon.com/guide/screenshot
Janryumon Mobile is, in my opinion, hands-down the best mobile Mahjong experience going. The graphics. The music. The animations. It's all done stunningly well and provides an video mahjong experience second to none. There are virtually no options to configure - fortunately the game uses fairly standard Riichi Mahjong rules (red 5s, Hidden Dora, etc). The offline play is free and the online play requires you pay out in game currency (GP) which replenishes slowly or you can pay for a big influx of game cash. I've been playing on the free GP handed out and have not needed to pay beyond that... but even with a pay-to-play model it works out very cheaply (300gp for a full Japanese regulation match and 12000GP costs US$7.99 so thats...um... roughly 20 cents per game).
Online you can choose 3P match or the normal 4P match. You can play a Quarter Game, Half Game (which is the Japanese full game) or just a single Round. They apparently have something like "friend codes" so you can play a friendly game with people you know.
Offline you have virtually no choices but you are treated to a very pleasing Japanese Riichi game. The animations of the hands coming in and drawing tiles, discarding them, putting them into the hand and pulling another to place out on the table is just fantastic. The music is equally great and kicks up a notch and changes tempo when there are less than 20 tiles or someone declares Riichi. Voices are fantastic and the scoring incantations at the end are fun. It's just highly polished. In the 4 months since I've been playing this particular game, they have released 2 updates via the App Store and appear to always be tweaking the experience and improving it (there is no cost on upgrades, obviously).
The interface is also quite nice - providing simple pop-ups and showing waits and highlighting similarly discarded tiles when you touch any tile. They have this natural motion detection on the iPad for calling and discarding so you can just learn to play without menus... though I still prefer the traditional pop-ups. They have this awesome little gleam that runs through the Dora tiles so you can't miss them. Everything about the presentation just oozes quality.

The game can be rendered into English or Japanese (or Korean? I think).
You can probably get a better sense of the game from the videos above but I've captured a few screenshots of my own (all in the native iPad resolution).





My wife Jenn is hooked on this mobile version. I was fortunate enough to steal away the iPad for 20 minutes to play a game and capture these screens. I got several dirty looks

Janryumon Mobile is highly recommended - I can't think of another 99 cent app (they have a low-resolution iPod trial version for free) that we've gotten more value from.