Before the 2.5 generation of cell phones mobile games came pre-installed with a device. If you wanted a new game you had to buy a new phone. Everything changed when mobile phones began to modernize. The introduction of Java ME, GPRS internet services and color display changed games. They became more advanced and could be downloaded on a phone via the mobile operator's network. Camera phones and EDGE internet services (called sometimes 2.75G) were another significant step in the evolution of mobile games. Their storage and graphic capabilities allowed production of higher quality games. Unfortunately Java games were still inferior in comparison with handheld console games due to the lack of processing power of the phones. Also there are a shitload of titles based on a movie licenses or dumbed down versions of console games produced only for quick cash. Because of that unlike in Japan mobile phone games never broke through in the US and Europe until the arrival of Apple and Android smartphones.
If you want to emulate Java games on your PC probably the best way will be KEmulator Lite In a case of issues with running it check Java Runtime Environment version installed on your PC. It should be ok with JRE 6 Update 33 but in a case of troubles try with other versions. There should be also Java ME emulators for iPhone/iPad and Android. If you want to play these games at your standard Java phone you can load them easily via Bluetooth, memory card or through WAP Games usually came in few different versions made for different phone models. They don't have to match exactly. A lot of games will run on different phone model that they were intended to. More important can be screen resolution or .jar file size so running them could be trial of error.
Java games are very easy to translate, there are a lot of tools for that on the internet. Also Japanese mobile games can't be emulated or played on non Japanese phones because of their distribution structure that resembles cloud gaming.
Death Trap II: The Unlocked Code
Adventure, Survival Horror
Side-scrolling adventure game somewhat similar to first Clock Tower. You must escape from an old mansion and find clues that evoke your lost memory. You fight monsters by setting traps.
Mr. Driller Deluxe
Mr. Driller, the Dig Dug spinoff about drilling through the Earth, is one of those games that feel perfect on a mobile phone. There are 500m, 1000m and Endless modes. All characters but the rabbit are available. Sadly, the controls aren't completely tight (a common issue with mobile games) and this can ruin playthroughs.
Mr. Last Minute
You are white-collar worker and your goal is to get to work on time. You compete against some stranger and dodge random generated hurdles that are on your way. Get tricks located on a levels and use them against your opponent like in some freaking Mario Kart game.
PK: Phantom Duck
The current last game about Donald Duck (make more already, Disney), which depicts his adventures as the superhero Paperinik. Has nice gameplay and animations.
Rayman Raving Rabbids
Instead of being a collection of minigames, this version is based on the original concept of Rayman 4, that is, Rayman battling crazy rabbits in a platformer. This version also happens to be a Sonic clone for some reason. Rayman rolls on slopes, goes on loops and finds springs and dash rings, too. The gameplay and level design are quite competent.
Silent Hill: Orphan (Silent Hill Mobile in Europe)
Adventure, Survival Horror
Non-canonical spin-off in the series. Plays like a first person point and click adventure game with some shooting sections. Puzzles are almost entirely logical and there is very little pixel hunting required. Very atmospheric. It was followed by two sequels that are also worth checking out.
Sonic The Hedgehog
A port of Sonic's first game, released in 2005 for Sega's Sonic Cafe japanese mobile service. It has less Blast Processing and lacks Special Stages, but was a better deal than that poor GBA port. When released in the west, it was sold in two parts.
Soul of Darkness
A decent Castlevania clone. You are vampire slayer who must rescue his girlfriend that had been kidnaped. It isn't Metroidvania but it borrows some of it's gameplay elements. You hack monsters and bosses, upgrade your weapons and use magic. It can be played with touch screen. There was remake for the DSiWare.
Super Action Hero
Platformer, Beat 'em up
As a badass stickman hero, you fight through various missions, kick ass with big combos to keep the timer from ending, and then unlock alternate faces to change your stats. Has a christmas spinoff, a direct sequel and another for the Zeebo.
Super Sketcher (Sketcher in Europe)
Very dynamic and one of most innovative mobile games. Sketcher is a hand-drawn character that runs like a crazy. You must draw missing parts of the levels to help him avoid hazards. In addition to arcade mode you can unlock bonus games that extend replayability.