Check on the Statue
After you defeat the final member of the Elite Four, he or she will tell you to check on the statue in the lobby. Go ahead and do that. You will be warped to the Champion's Room.
When you get to the Champion's room, you find none other than N facing off with Alder. In fact, N has just defeated Alder. Then, after some dialog, N will make his castle rise from the ground. Its walls will surround the Pokémon League, and a bunch of staircases will shoot out into the Pokémon League.
N departs for his castle, and then Cheren arrives. He and Alder urge you to go into the castle, so do that.
When you get into N's Castle, six of the Seven Sages approach you. There's no way you could take all of them on! Luckily, a bunch of the Unova Gym Leaders arrive on the scene and team up against the sages, giving you the chance to puruse N.
N's Castle 2F
On this floor, you will find Anthea and Concordia, who appeared in the intro sequence of the game. Anthea will heal up your pokémon for free. In a room on this floor, you will find a Max Potion.
N's Castle 3F
There is a PC in a room on this floor. There is also a Max Revive in one of the rooms, and there is a grunt who will give you an Ultra Ball. In addition, a grunt will ask you if you want to return to the Pokémon League. If you say yes, you will be warped into the Pokémon Center of the Pokémon League. A grunt there will warp you back to the castle. This means that if you have any preparations that you want to make before this next part, you can do so.
N's Castle 4F
On this floor, you will find N's room, where he was cut off from other people and shown only pokémon who had been hurt by humans. There is a Rare Candy here. In another room on this floor is a Full Restore.
N's Castle 5F
Ghetsis will step out of the throne room and talk to you for a little while. Then you can go into the throne room.
N's Throne Room
N will talk for a while. Walk toward N when you can. Eventually he will summon his legendary dragon, which will crash through the wall behind N's throne. Then your Light/Dark stone will float up and react, and it will transform into the other legendary dragon.
As soon as you talk to the legendary dragon, you will enter into battle with it. You have to catch the dragon in order to proceed, unless your PC and party are both full. Be careful not to talk to the dragon until you're ready to try to catch it. Save the game first if you wish, then go and do whatever training you want to do. You can come back and talk to the dragon whenever you're ready.
Catching Reshiram or Zekrom
The game programmers were somewhat merciful in that they gave Reshiram and Zekrom a higher catch rate than most legendary pokémon. Presumably they didn't want it to be too frustrating to try to catch a pokémon that you are required to get in order to proceed.
You should use False Swipe on the dragon if you can. However, a bunch of the pokémon who can learn False Swipe are weak to the dragon's moves. Read about how to get False Swipe.
Also, if possible, inflict Sleep or Freeze on the dragon before trying to catch it. If this is not possible, try inflicting Paralysis.
Then, throw Dusk Balls if you have them. Otherwise, use Timer Balls if 30 turns have passed, or if all else fails, throw Ultra Balls.
If you KO the dragon, or your party is defeated, the dragon will still be standing there waiting for you to catch it, so you can try as many times as you need to.
Once you catch it, you can choose to send it to your PC or instead send one of your current pokémon to your PC so as to have your legendary dragon with you for the final battles. It would be wise to keep the legendary dragon with you, because it's really tough.
After you catch the legendary dragon, there are a couple of battles that you will need to do, and you won't be able to save the game until after you have won, so be prepared.
Final Showdown with N
Once you have caught the legendary dragon of your game, N will heal up your pokémon and challenge you with his six pokémon, which includes his legendary dragon.
In Pokémon Black, his Level 52 Zekrom has the Teravolt ability, which nullifies the following abilities: Battle Armor, Clear Body, Damp, Dry Skin, Filter, Flash Fire, Flower Gift, Heatproof, Hyper Cutter, Immunity, Inner Focus, Insomnia, Keen Eye, Leaf Guard, Levitate, Lightningrod, Limber, Magma Armor, Marvel Scale, Motor Drive, Oblivious, Own Tempo, Sand Veil, Shell Armor, Shield Dust, Simple, Snow Cloak, Solid Rock, Soundproof, Sticky Hold, Storm Drain, Sturdy, Suction Cups, Tangled Feet, Thick Fat, Unaware, Vital Spirit, Volt Absorb, Water Absorb, Water Veil, White Smoke, and Wonder Guard.
N's Zekrom knows Fusion Bolt (Electric Physical), Zen Headbutt (Psychic Physical), Giga Impact (Normal Physical), and Light Screen (Psychic Status). Being a Dragon/Electric pokémon, Zekrom is weak to Ground, Ice, and Dragon moves.
If you are fighting Zekrom and you have Reshiram in your party, try attacking right away with DragonBreath. You might be able to withstand Zekrom's attacks to defeat it with Reshiram even though it's weak to Zekrom's DragonBreath. Otherwise, Gigalith with Earthquake or Bulldoze might be a good opponent, or Crustle with Earthquake or Bulldoze.
In Pokémon White, N's level 52 Reshiram has the Turboblaze ability, which has the same effect as Teravolt (listed above).
N's Reshiram knows Hyper Beam (Normal Special), Fusion Flare (Fire Special), Extrasensory (Psychic Special), and Reflect (Psychic Status). Being a Dragon/Fire pokémon, Reshiram is weak to Ground, Rock, and Dragon moves.
If you are fighting Reshiram and you have Zekrom in your party, DragonBreath and Zen Headbutt might be enough to take it down. If not, try sending in Scrafty with Dragon Claw, Dragon Tail or Rock Slide, Gothitelle with Rock Slide, or Darmanitan with Earthquake, Bulldoze, or Rock Slide.
As for the rest of N's party, he has an L50 Klinklang that knows Flash Cannon (Steel Special), Metal Sound (Steel Status), Hyper Beam (Normal Special), and Thunderbolt (Electric Special). It is weak to Fighting, Ground, and Fire moves.
He also has an L50 Zoroark that knows Flamethrower (Fire Special), Focus Blast (Fighting Special), Night Slash (Dark Special), and Retaliate (Normal Special). This Zoroark has the Illusion ability, so it will look like a different pokémon from N's party. Pay attention the moves that it uses to see if you can tell if it's Zoroark instead of something else. Zoroark is weak to Bug and Fighting moves.
N also has an L50 Carracosta that knows Stone Edge (Rock Physical), Aqua Jet (Water Physical), Crunch (Dark Physical), and Waterfall (Water Physical). It is super-weak to Grass, and weak to Fighting, Ground, and Electric moves.
Also, N has an L50 Archeops that knows Stone Edge (Rock Physical), Acrobatics (Flying Physical), Dragon Claw (Dragon Physical), and Crunch (Dark Physical). It is weak to Rock, Steel, Water, Electric, and Ice moves.
Finally, N has an L50 Vanilluxe that knows Blizzard (Ice Special), Hail (Ice Status), Flash Cannon (Steel Special), and Frost Breath (Ice Special). It is weak to Steel, Fighting, Fire, and Rock moves.
Final Battle with Ghetsis
After you defeat N, Ghetsis will barge in and express his disappointment with N, who turns out to be Ghetsis's son. Ghetsis admits that he deliberately raised N to be isolated from humans, with only pokémon as friends, in order to manipulate N into being the king of Team Plasma who could capture the legendary dragon and make everyone release their pokémon. Ghetsis's plan all along was to become the only person in the region who had any pokémon so that he could take over and be in control. In order to try to protect his plan, he battles you.
Ghetsis has an L52 Cofagrigus that knows Toxic (Poison Status), Shadow Ball (Ghost Special), Protect (Normal Status), and Psychic (Psychic Special). It is weak to Ghost and Dark moves.
He also has an L52 Bouffalant that knows Head Charge (Normal Physical), Wild Charge (Electric Physical), Earthquake (Ground Physical), and Poison Jab (Poison Physical). It is weak to Fighting moves.
He also has an L52 Seismitoad that knows Muddy Water (Water Special), Rain Dance (Water Status), Earthquake (Ground Physical), and Sludge Wave (Poison Special). It is super-weak to Grass moves.
Ghetsis also has an L52 Bisharp that knows Night Slash (Dark Physical), Stone Edge (Rock Physical), X-Scissor (Bug Physical), and Metal Burst (Steel Physical). It is super-weak to Fighting moves, and is weak to Ground and Fire moves.
Also, Ghetsis has an L52 Eelektross that knows Crunch (Dark Physical), Flamethrower (Fire Special), Acrobatics (Flying Physical) and Wild Charge (Electric Physical). Eelektross has no weaknesses.
Finally, Ghetsis has an L54 Hydreigon that knows Dragon Pulse (Dragon Special), Fire Blast (Fire Special), Surf (Water Special), and Focus Blast (Fighting Special). It is weak to Fighting, Bug, Ice, and Dragon moves.
After N and Ghetsis
I won't spoil the ending for you! Just watch it! Be sure to wait through the end credits, because the game will automatically save afterward (you'll have to press a button when The End appears on the screen). After it saves, it sends you back to the title screen of the game.