To unlock a certain character's emblem, beat ten missions with that character. Unlock the following emblems by performing the corresponding tasks:

EMBLEM - HOW TO UNLOCK
Aerial Ace Emblem - Broke the 2000.000m record in Rocket Ski Jumping.
Amy Emblem - Complete 10 missions with Amy.
Backcountry Emblem - Finish Cross Country in 14:00.00 or less.
Blaze Emblem - Complete 10 missions with Blaze.
Blizland Emblem - Get gold/silver stars on all Blizland missions.
Blizza Emblem - Rescued Blizza the snow spirit.
Bowser Emblem - Complete 10 missions with Bowser.
Bowser Jr. Emblem - Complete 10 mssions with Bowser Jr.
Broom Master Emblem - Get three points or more in one end in Curling.
Collaborative Emblem - Played as all characters.
Cuby Emblem - Rescued Cuby the snow spirit.
Cubyrinth Emblem - Get gold/silver stars on all Cubyrinth missions.
Daisy Emblem - Complete 10 missions with Daisy.
DK Emblem - Complete 10 missions with DK.
Dr. Eggman Emblem - Complete 10 missions with Dr. Eggman.
Dream Dancer Emblem - Got more than 3000pts in Ultimate Firgure Skating.
Drift Boarder Emblem - Finish Snowboard Cross in 55.00 seconds or less.
Drift Monster Emblem - Finish Extreme Snowboarding in under 3:00.00.
Explorer Emblem - Complete Adventure Tours.
Express Emblem - Finish Bobsleigh in 57.00 seconds or less.
Friendship Emblem - Played in Single Card play or Multi Card play mode.
Frost Emblem - Rescued Frosty the snow spirit.
Frostown Emblem - Get gold/silver stars on all Frostown missions.
Full Play Emblem - Played all events.
Gate Master Emblem - Finish Alpine Skiing Gs in 50.00 seconds or less.
Ghost Fighter Emblem - Defeat all Level 2 Ghosts.
Goal Hunter Emblem - Scored more than 100pts in Fever Hockey.
Good Evening Emblem - Started the game between 6:00PM and 6:30PM.
Good Morning Emblem - Started the game between 6:00AM and 6:30AM.
Ground Search Emblem - Found a Whitestone in the ground.
High Speed Emblem - Finished Luge in 00:57.000 or less.
High Speed Emblem - Finish Luge in 50.00 seconds or less.
Ice Rink Dancer Emblem - Got more than 100.000 pts in Figure Skating.
Ice Warrior Emblem - Won in Ice Hockey by a 10-point margin.
Icepeak Emblem - Get gold/silver stars on all Icepeak missions.
Icy Emblem - Rescued Icy the snow spirit.
Item Collector Emblem - Collected all Items.
King of Skiers Emblem - Finish Nordic Combined in 14:30.00 or less.
Knuckles Emblem - Complete 10 missions with Knuckles.
Long Jump Emblem - Jumped Beyond 133.000m in Ski Jumping LH.
Luigi Emblem - Complete 10 missions with Luigi.
Mario Emblem - Complete 10 missions with mario.
Metal Sonic Emblem - Complete 10 missions with Metal Sonic.
Mini-Master Emblem - Completed all minigames in minigame play.
Network Emblem - Connected to Nintendo WFC Rankings.
Non Stop Emblem - Finished Skeleton in 00:57.000 or less.
Party Emblem - Win in each of the Party Games.
Peach Emblem - Complete 10 missions with Peach.
Point Hound Emblem - Get at least 5 badges in Snow Machine Fight.
Pola Emblem - Rescued Pola the snow spirit.
Polastraits Emblem - Get gold/silver stars on all Polastraits missions.
Round Champion Emblem - Win a Multi-Round match.
Shadow Emblem - Complete 10 missions with Shadow.
Sharpshooter Emblem - Get more than 15,000 points in Ski Shooting.
Silver Emblem - Complete 10 missions with Silver.
Snow Shooter Emblem - Finish Biathlon in 15:00.00 or less.
Sonic Emblem - Complete 10 missions with Sonic.
Souvenir Collector Emblem - Collect all the Olympic Memorabilia.
Sparkleton Emblem - Get gold/silver stars on all Sparkleton missions.
Sparky Emblem - Rescued Sparky the snow spirit.
Speed Demon Emblem - Finish Intense Short Track in under 2:00.00.
Speedster Emblem - Finish Supersonic Downhill in 21.00 seconds or less.
Spot Search Emblem - Found a Whitestone hidden in a barrel or a crate.
Sprinter Emblem - Finished Speed Skating 500m in 00:33.000 or less.
Tails Emblem - Complete 10 missions with Tails.
Terrain Racer Emblem - Finish Ski Cross Racing in 1:30.00 or less.
Thanks for Playing Emblem - Started the game 50 times.
Track Skater Emblem - Finished Short Track 500m in 00:40.000 or less.
Traveler Emblem - Rode all Gondolas.
Trivia Collector Emblem - Collect all the Trivia.
Vector Emblem - Complete 10 missions with Vector.
Waluigi Emblem - Complete 10 missions with Waluigi.
Wario Emblem - Complete 10 missions with Wario.
Yellow Bib Emblem - Got more than 25,000pts in Moguls.
Yoshi Emblem - Complete 10 missions with Yoshi.

Unlock the following villains by performing the corresponding tasks:

VILLAINS - HOW TO UNLOCK
Bowser and Eggman - After unlocking the two above, do the last battle with Bowser and Eggman again (in the main game mode, not play mission mode).
Bowser Jr. And Metal Sonic - Finish all of the new missions (the green ones) the two put out.