The cube is made with two-colour archetypes in mind but the summaries below are only suggestions. Three-colour or more decks are definitely do-able and there are different sub-themes throughout, like artifacts, enchantments and landfalls, to explore.
Azorius (Blue/White)
UW has two main themes; blink and flying. Blink decks use cards like Ephemerate to re-trigger ‘enter the battlefield’ effects on creatures like Cloudgoat Ranger and Inspiring Overseer. Decks with lots of fliers look to use evasion to get past the opposition’s blockers.
Dimir (Blue/Black)
U/B is focused on control with plentiful removal and counterspells. You might delay your opponents with efficient black removal cards like Doom Blade or blue counter spells like… Counterspell. Then when they run out of steam, you can refuel your hand with draw spells like Opt to make sure you find your win conditions.
Rakdos (Black/Red)
B/R decks have access to plentiful removal and a sacrifice theme. You can generate sacrifice outlets with spells like Hordeling Outburst or by taking control of your opponents’ creatures with cards like Vengeful Possession.
Gruul (Red/Green)
R/G decks are home to the cube’s biggest beasts. Cards like Drowsing Tyrannodon want you to play your threats with power 4 or above and there are plenty of them - like Earthshaker Dreadmaw and Charging Monstrosaur. Once your threats are in play, cards like Colossal Majesty will make sure you’re drawing new cards to replace those your opponents inevitably focus on with their removal.
Selesnya (White/Green)
In W/G decks, you’ll be using cards like Forced Adaptation to place +1/+1 tokens on creatures like Basking Broodscale that may not look like much at first but reward you as they get bigger. Cards like Abzan Falconer give a boost to any of your creatures with a +1/+1 token.
Orzhov (White/Black)
W/B decks look to slowly drain their opponent’s life total and boost their own by sacrificing small creatures. Cards like Lingering Souls create a persistent supply of sacrifice fodder to feed into payoffs like Bastion of Remembrance.
Izzet (Blue/Red)
U/R decks can have access to a lot of instants and sorceries and also has payoffs for playing them. Blue has cheap card draw like Thought Scour and counter magic like Essence Scatter while Red has access to efficient burn and removal spells like Lightning Bolt. U/R decks can get a range of benefits from playing spells (for instance cards with Prowess like Jeskai Elder) and once the graveyard is full of instants and sorceries, big hitters like Tolarian Terror become much more affordable.
Golgari (Black/Green)
B/G decks can get value from the graveyard. The focus here is on getting value from cards like Chalk Outline and recurring your threats with cards like Reanimate rather than cheating out big creatures early, though there are one or two ways to do that if you can bring it together.
Boros (White/Red)
Cards like Resolute Reinforcements and Goblin Instigator give W/R decks lots of ways to to wide with tokens. From there the deck can go in different directions. Cards like Rally the Peasants give a team-wide buff while others like Impact Tremors make token generation an aim in itself.
Simic (Blue/Green)
B/G decks have an abundance of ways to ramp your mana with cards like Arboreal Grazer and Llanowar Elves. Mana dorks and fast land help you rush out costly big creatures while blue card draw spells help make sure you always have something good to be playing.