Control is a family of decks that are built to last for many, many turns, completely control the game (hence the name) and end it with a single finisher, which is usually a creature but may be a repeatable effect or big instant, sorcery or ability as well.
Cube Decklist: UB Control
Constructed Decklist: Psychatog
Relative popularity in cube: Common
Perhaps the most classical combination of colors for control decks, UB combines blue's counterspells, card drawing and versatility with black's efficiency at creature removal and hand disruption. Those two colors, together, can attack the opponent's plans through pretty much any angle. For this archetype to work, it is important to have a deck with answers to everything, so counterspells are a must. Creature removal, discard, finishers and card drawing complete the core of UB Control. Card advantage is very common in these two colors, so typically there will be a lot of options available to overwhelm the opponent as the game goes on.
This is a powerful and easy to draft archetype, making it a good option when blue and black are open. What it's good against and bad against varies too much from build to build. It can be weak to aggro if it doesn't have enough removal, to midrange if it doesn't have enough counterspells, to aggro-control if it doesn't have enough discard or to other control if it doesn't have enough card drawing. The key is making a balanced build that can do well against all sorts of decks.
Rare Variation: UB Milling
A twist on UB Control is UB Milling, which works the same way, except it doesn't use big creatures as finishers, but milling. As games typically last at least 15 turns with UB Control, this means milling 20 cards from the opponent while controlling him is enough to win. This plan is typically slower than using fatties, but more reliable because there isn't much for the opponent to do about it except win before he is decked.
To make UB Milling work, simply replace finishers with milling cards. Archive Trap is very efficient, and even Millstone does alright in limited. Assume the game will last 13 turns, which added to the 7 cards in the opponent's starting hand makes his "life total" regarding milling around 20. Archive Trap would therefore be a burn that deals 13 damage, and Millstone would deal 2 damage a turn for 2 mana each activation.
Milling also has synergy with reanimation that can take creatures from your opponent's graveyard. While they will attack a different dimension than the rest of your deck, it'll be invaluable for defense and using utility creatures that end up there, or eventually some big creature that can take the game by itself.
Cube Decklist: WU Control
Constructed Decklist: Mirari's Wake
Relative popularity in cube: Common
Blue's main weakness is its inability to deal permanently with creatures, and that's why monoblue control decks are nearly impossible to be successful with. Black solves the problem with its removal, but so does white. In fact, white is better than black to deal with creatures - its removal is generally more powerful - see Swords to Plowshares - and it has access to several mass removal spells. Also, it adds the important ability to remove enchantments and artifacts that are already on the battlefield. Lastly, white has good defensive creatures and life gain, making it a great color to stop aggressive decks.
The downside in comparison with black is the lack of discard, which increases the importance of counterspells. Another disadvantage is that white does not generate as much card advantage as black does, so on average it's a better deck to deal with beatdown, but worse in the control matchup.
Rare Variation: WU Aggro-Control
This not-so-great control matchup causes WU Control to want to kill the opponent as soon as the situation becomes favorable. One good option is using evasive creatures and protecting them from the start of the game, and focusing more on tempo control than mass removal, while retaining the counterspells, creature pinpoint removal and enchantment/artifact removal. This creates an aggro-control build that deploys clocks early, at the expense or long term power. It's a delicate build to brew, which can only work with good evasive creatures, but quite effective when done right.
There is also the option of using this as a plan B with a transformative sideboard. When WU Control faces another slow, grindy deck, it can become WU Aggro-Control by replacing mass removal and expensive finishers with cheap and prefarably evasive creatures.
Cube Decklist: UR Control
Constructed Decklist: Counter-Burn
Relative popularity in cube: Rare
Lastly, blue can combine with red, which has access to tons of creature removal, creating the archetype known as Counter-Burn. While it does not offer discard and card advantage like black does, or versatile removal and defense like white does, red is very, very efficient at killing creatures and artifacts, making it even better than white against aggro. Another plus is that expensive burn spells can be used as finishers, combined or not with fatties - a slower but harder to stop win condition, analogous to milling.
It's easy to draft burn for UR Control since there is so much of it in the cube, but an important detail is to prioritize expensive burn, like X spells or recurrent ones like Pulse of the Forge and Hammer of Bogardan. The blue part should have the usual - counterspells, card selection, and more importantly than in other archetypes, card drawing. Burning has the capacity of turning card advantage into victory quickly - there are few cards you would rather draw a lot of than Lightning Bolts.
Cube Decklist: WB Control
Constructed Decklist: Decree
Relative popularity in cube: Rare
A control deck without blue will miss its counterspells and card drawing, which are key elements in the archetypes described before. White and black can, together, fill these holes partially with black's card advantage and discard and white's removal versatility. Since both colors are great at killing creatures, this won't be an issue at all for WB Control, so its aggro and midrange match ups are pretty good. It suffers, however, against decks that are light on creatures and run removal, like control and aggro-control. The only mechanism to protect your finishers will be discard, which is unreliable near the end of the game.
It is important to avoid putting too much creature removal in WB Control. It's easy to fall for this trap when drafting WB, but overloading your deck with Doom Blades and Journey to Nowheres will make it very bad against creature-light decks, especially control. It is preferable to fill some of those spots with resilient win conditions, even if they are suboptimal. You will have time to cast expensive spells, but remember you won't have a grip of counterspells to make sure they stick.