In order to battle Red in HeartGold/SoulSilver, you must have beaten all 16 Gym Leaders of Kanto and Johto, get Rock Climb, and go through Mt. Silver Cave. Red's Pikachu is level 88, Lapras is level 80, Blastoise is level 84, Venusaur is level 84, Snorlax is level 82, and Charizard is level 84.
While fighting Red it will start to Hail, unless you have a weather effect Pokémon such as Rayquaza. When fighting Red, he will send out Pikachu first, and he will use Volt Tackle, unless yours is a different type, it will be Iron Tail. After beating Red, he will disappear, then the game will end as if you had beaten the Elite Four. Red does not disappear forever, after you beat the Elite Four again, Red will be at Mt. Silver again. This location first appeared in the original Gold and Silver games.
Also in Mt. Silver, Moltres resides there at level 50. In order to get Moltres, you need to have a Pokémon with Rock Climb in your party and also a Pokémon who knows Waterfall.
In the series
In the series, Ash, Misty, and Brock travel to the mountain to drop off Larvitar and enter the Johto League Silver Conference. The Silver Conference is at the foot of Mt. Silver.