The Ace: Seemingly as smart as Lelouch but as athletic as Suzaku, Rai excels at virtually everything he does. He's far from unbeatable or perfect, though (for example, in the Black Knights route he's handily defeated by Guilford and later overwhelmed by superior numbers despite his own Super Prototype).
Ace Custom: In the routes where he becomes a Knightmare pilot, Rai initially starts out with a standard model of whichever faction he joins, but he quickly receives an improved custom version before finally upgrading to a Super Prototype.
Compelling Voice: Compared to Lelouch's Geass (which depends on eye contact as well as a vocal command), Rai's Geass works on sound alone. This means that he can use his Geass even without directly seeing whoever he wants to command (for example, he orders a group of terrorists, and only the terrorists, to commit suicide).
Dark and Troubled Past: Rai was actually a member of the Britannian nobility, though like Kallen he had mixed heritage. Specifically, his mother was Japanese, which of course led to him and his family being ostracized. When V.V. came around, he offered him the power of Geass. Rai accepted and used his Geass to kill his Britannian family in order to gain control of the land he was given dominion over. Unfortunately, his happy life didn't last very long as he didn't quite understand everything about his Geass and, like Lelouch's mistake with Euphie, he ordered everyone to fight for their deaths when his home was invaded. By the time the smoke cleared, everyone he cared about was gone and was Driven to Suicide. V.V., however, didn't let him as he hadn't fulfilled his end of their bargain and locked him away inside a Thought Elevator.
Defrosting Ice Queen: Goes through this as the game progresses. Some characters point it out with lines like, "That's the first time I've seen you laugh," or "So you can smile after all."
Despair Event Horizon: Crossed this in the past, as his Geass going out of control resulted in the deaths of his mother and little sister, the two people he wanted to protect the most.
From Nobody to Nightmare: Has shades of this in the Brittanian Army route, especially if Rai joins Princess Cornelia's Royal Guard. He basically comes from nowhere and yet in the space of a few weeks has gone from a new recruit hired as part of an experimental unit (as a test pilot, no less) to a member of Cornelia's elite Royal Guards.
Likewise, in the Black Knight route he can go from a new recruit to Zero's right hand man.
Infinity -1 Sword: The special Knightmare Frames that Rai gets to pilot for the Black Knights or Brittanian Army, the Prototype Gekka and Lancelot Club respectively, are this. They lack the energy efficiency and power output of the Guren and Lancelot due to being built with spare parts, but they are still stronger than the average Knightmare Frame, and only Rai is capable of piloting them.
Part of this aspect of his personality comes from the fact he's amnesic, while another part comes from a fear of losing people close to him again.
Player Character: He's the protagonist of the game, whose personality can be shaped by the player's choices.
Pretty Freeloader: Like Lelouch and Nunnally, Rai is staying in the school clubhouse thanks to Milly being one of the Ashfords and thus having a lot of pull at the school. Rai at least feels guilt about this (despite Milly's assurances) and helps out at the student council to make up for it.
Royals Who Actually Do Something: Rai turns out to be half-Brittanian and half-Japanese, all royal. On his father's side, he is related to the Brittanian Royal Family (and apparently even ruled the Brittanian homeland briefly) while on his mother's side he is related to the Japanese Imperial Family (similar to Kaguya Sumeragi, leader of the Kyoto Six).
Self-Made Orphan: He killed his own father and half-siblings for mistreating his mother and little sister. He later accidentally kills his mother and little sister as well when his Geass goes out of control and they march to their deaths in battle alongside his army and every civilian who heard his voice.
The Stoic: Several characters (including Milly and Suzaku) observe that he doesn't show much emotion, at least at first.
Tropes specific to Rai in the Geass route
The Call Knows Where You Live: C.C. senses his Geass and eventually recruits him into the Black Knights, despite his own uncertainty about the situation.
Despair Event Horizon: Crosses this when his memories return and he realizes his Geass is going berserk again.
Incurable Cough of Death: Claims he's undergoing this, which he uses as an excuse to leave the Black Knights. Lelouch is the only one that knows he's lying, but says nothing as he respects his decision. Afterwards, Rai disappears and seals himself inside the Thought Elevator again, going so far as use his Geass to erase everyone's memories of him.
Tropes specific to Rai in the Order of the Black Knights route
The Dragon: Can become this to Zero, taking the place of Ohgi from the Code Geass canon.
Secret Keeper: In one possible end of the Black Knights Route, Rai is one of the few people who knows that Lelouch is Zero, on account of Lelouch revealing himself to him of his own volition.
Tropes specific to Rai in the Brittanian Army route
Bodyguard Crush: Should he become one of Cornelia's Royal Guards, it's hinted that Cornelia might have a (very) slight one on Rai (she actually shows some interest in his well-being and progress, which some people who know her find unusual).
Meanwhile a lot of other characters wonder if Rai has one of these for Cornelia. Part of this is because Rai often seeks her out to speak to her (despite their conversations almost always ending with her scolding him). Rai himself lampshades this with the thought, "Ah. She got mad again."
Driven to Suicide: Unlike other routes where his Geass begins going out of control, Rai actively tries to get himself killed in the Royal Guards ending.
Tropes specific to Rai in the Ashford Academy route
Heroic BSoD: When forced at the end of the route to use his Geass to compel attacking terrorists to commit suicide (in front of his friends and other witnesses, no less), Rai has one of these and quickly commands everyone present to forget what happened before fleeing into the night and vanishing forever.
Mysterious Waif: Rivalz informs Rai that he's developed quite the following in school due to being seen as one of these. He even has a nickname: The 'Illusionary Adonis'. Rai himself is surprised and more than a bit embarrassed to learn this.
Tropes specific to Rai in the Japanese Liberation Front route (available in the PSP version only)
Back from the Brink: Rai's involvement and opposition to Zero helps the JLF make a comeback from almost being destroyed by Cornelia's forces to a viable alternative to the Black Knights as the premiere Japanese resistance movement.
Everyone Has Standards: When Rai realises that Zero intends to sacrifice the JLF in order to accomplish his objectives, he ignores his orders and goes to assist them. This is caused by his outrage that the JLF are to be sacrificed without their knowledge.
Spanner in the Works: Rai unknowingly spoils Zero's plans at least twice: once by going to the aid of JLF forces during the battle at Narita (which allows a number of JLF forces who would've otherwise been destroyed to escape) and another during the battle over the tanker serving as JLF HQ. In the anime, when the tanker is boarded by Brittanian forces Zero detonates a bomb that destroys the tanker, then claims the JLF leadership chose death over capture. Rai is able to disable the boarding forces, thus robbing Zero of his chance to remove the JLF as political rivals.
Tastes Like Purple: Kaguya explains to Rai she was the one who ordered his customised Gekka to be coloured blue, telling him that he "feels blue." She elaborates that he reminds her of the sea, both calm and fierce.
Tropes specific to Rai in the Blue Moon route (available in the PS2 version only)
No Good Deed Goes Unpunished: Everything Rai does seems to just piss Kaguya off more and more, despite his best efforts to make amends. This is especially jarring if you've played the JLF route, where Kaguya develops a bit of a crush on him.