Here's a link to the Official Article on Conjunctions.

Conjunctions are fellowship combat maneuvers that can deal significant damage or restore power or morale. The more powerful ones can combine these, affect multiple targets, or add heal-over-time buffs on you. Some even summon creatures to help you battle!

Conjunctions are started randomly when you're fighting signature or elite mobs. Some classes (such as burglars) have skills or items that can start (also known as opening) a conjunction.

Once a conjunction starts, the mob is stunned, and a conjunction wheel will appear on the screen of each fellowship member who is targeting that mob. (Those not targeting it will see a bulls-eye, clicking that will allow them to switch targets to the appropriate mob, and the wheel will open.) The wheel has four colored circles: RED (which does damage), YELLOW (which does damage over time), GREEN (which restores morale), and BLUE (which restores power).

Each player has a few seconds to select one of the colors. Once you choose a color, it will appear in the Conjunction Contribution Panel (so everyone can see what has been chosen). This freezes your character (though other mobs can still attack you). If you move, your contribution is removed. After a few seconds (whether you choose a color or not), the conjunction resolves.

Some contributions require proximity to the mob, others can be used at range. There are slightly different icons on the conjunction wheel that indicate the range; if you look closely you'll see some arrows on the RED and YELLOW circles on the Hunter's wheel, for example. We think BLUE and GREEN can be used at range for all classes. We're still testing how this works, and will fill out this section as we learn more:

• Burglar - melee: RED/YELLOW
• Captain
• Champion
• Guardian
• Hunter - all ranged
• Lore-master
• Minstrel - melee: RED/YELLOW

Here's a video showing three different (length-three "straight") conjunctions: http://www.youtube.com/watch?v=2e5p-B68Tik

Notice the effect on the mob's health, and on the players' morale and power.

Although every conjunction with at least one color does something, there are special combinations which are extremely powerful. These tend to work like poker hands (pairs, flushes, straights), and are described in detail below. Straights describe a sequence that goes in order (clockwise or counter-clockwise) around the conjunction wheel. (Note: The wheel for Hunters is different; this is generally considered to be a bug.)

For the pairs like RRYY, the reverse order is equivalent. Only one order is listed here. That is not true for full-houses though; that is XXXYY is not the same as YYXXX.

Although I've completed all of the length 5 and 6 flushes, they do not appear in my Fellowship Skill list, so I assume they do not have special names.

Here I transcribed all the ones I've completed, along with the in-game text. The ones with question marks I have not yet personally completed. The names come from other posts. For "straights" (i.e., working your way around the conjunction wheel), I believe that the direction (clockwise or counter-clockwise) does not matter. I know that's true for the length-four ones, but I have not confirmed it for length-five or six.

ONE LONG

R - Ent's Strength 105% of Main hand weapon + 328 damage. (effects vary by level; we're working on the formulas)
Y - Spider's Guile Does 130 common damage every 4 seconds for 20 seconds.
B - Stallion's Spirit Adds 268 - 274 to Power.
G - Eagle's Cry Adds 434 - 439 to Morale.

LENGTH TWO FLUSHES

RR - Clash of Arms You and your fellow smash your target as if caught between a hammer and anvil.
YY - Misdirection Working together with a fellow you both are able to cause deep wounds in your target.
BB - Planned Attack Your superior coordination with your fellow energizes you both.
GG - Rallying Cry You and your fellow let out a piercing rallying cry invigorating yourselves and all other fellowship members around you.

LENGTH THREE FLUSHES

RRR - Resounding Strikes Striking together you and your fellows tear your target asunder.
YYY - Deception Working together with your fellows you all are able to cause deep wounds in your target. The initial contributor will cause a deep wound.
BBB - Three Pronged Assault Your superior coordination with your fellows energizes you all.
GGG - Mustering the Troops You and your fellows let out a piercing rallying cry invigorating yourselves and all other fellowship members around you.

LENGTH THREE STRAIGHTS

RBY - Sinister Plan Working with your fellows you are able to slash and wound your target while also energizing your fellowship members.
RGY - Wrath Of the Righteous Working with your fellows you are able to crush and wound your target while also rallying your fellowship members.
GRB - Strength of the Pure Working with your fellows you are able to crush your target while also rallying and energizing your fellowship members.
GYB - Will of the Strong Working with your fellows you are able to wound your target while also rallying and energizing your fellowship members.

LENGTH FOUR FLUSHES

RRRR - Hail of Blows Striking together you and your fellows tear your target asunder.
BBBB - Power In Numbers Your superior coordination with your fellows energizes you all.
GGGG - Call To Arms You and your fellows let out a piercing rallying cry invigorating yourselves and all other fellowship members around you.
YYYY - Chaos in the Ranks Working together with your fellows you all are able to cause deep wounds in your target. The initial contributor will cause a particularly nasty wound.

PAIRS

RRYY - Pounding Echos Your fellowship's pounding blows echo in the distance and your foes new wounds are great.
RRBB - Planned Charge With a well planned charge, your fellowship crushes the enemy while being energized.
RRGG - Ent's Heart With an Ent's resilience and strength, your fellowship crushes your target as they rally back.
YYBB - Deadly Whispers Your fellowship's deadly whispers herald a deep wound in your foe and an energizing triumph.
YYGG - Silent Glory Your fellowships silent wound and worthy glory rally everyone.
BBGG - Noble Lineage Like ancient nobles, your fellowship is able to crush your target while also rallying everyone's morale.

LENGTH FOUR STRAIGHTS

RGYB or RBYG - Ent's Stand Like an Ent pooling all resources, your fellowship can crush and wound your target while also rallying and energizing all.
BRGY or BYGR - Noble's Honour With a true noble's honour your fellowship is able to crush and wound your target while also rallying and energizing all.
YBRG or YGRB - Silent Paths Working with your fellows you are able to crush and wound your target while also rallying and energizing your fellowship members.
GYBR or GRBY - Eagle's Eyrie The power of great eagles in their eyries is matched by your fellowship as you crush and wound your target while rallying and energizing all.

FULL HOUSE

GGGBB - Pure of Heart The pure of heart fear no foe and your fellowship can rally and energize in the midst of any battle.
BBBGG - Noble Blood Like those of true noble blood, your fellowship rallies and energizes from all past loss.
YYYRR - Whispering Leaves Working with your fellows you are able to crush and wound your target.
RRRYY - Ent's Rage With an Ent's strength and with your fellowship's blows, crush and wound your foe.

LENGTH FIVE STRAIGHTS

BRGYB or BYGRB - Wrath of the Oathbreakers Your fellowship strikes with such wounding wrath as to summon an Oathbreaker to assist you even as you rally and energize all.
RGYBR or RBYGR - Entish Justice Justice of the Ents does not rest with a single foe, all enemies feel your fellowships crushing and wounding rage even as you rally and energize each ... (unreadable)
GYBRG or GRBYG - Valour Unashamed Your fellowship crushes and wounds your target with unashamed valour, which removes fear, restores morale and enegizes all.
YBRGY or YGRBY - Leaves of Lothlorien Swirling like the leaves of Lothlorien, your fellowship strikes from all sides, leaving wounds that slow while rallying and energizing all.

LENGTH SIX UNIQUE

RRRYYY - Tramp of Doom Your fellowship tramples your foes, inflicting deep wounds.
RRRRRG - Roll the Drum The rolling drum of your fellowships blows on your foe cause pain to nearby foes and hope to flourish in your ranks, removing fear and wounds.
RRRRRY - Break the Door You fellowships fury cripples your foe and crushes all nearby enemies.
RRRRRB - Hew the Stone Your fellows hew into your foes like masons on stone, energizing each other and drawing forth an Oathbreaker shade to fight for you.
BRYYRB - Breath of Freedom Your fellowship strikes with the conviction of freedom, defeating your foes and rallying your allies.

GRRRRB - March of the Ents ?
GYYYYB - Chill of Bone Your fellowship strikes deep, bone chilling wounds on your foe, rallying and energizing each other as they fight.

LENGTH SIX STRAIGHTS

BRGYBR or RBYGRB - Wings of the Windlord A truly flawless display of fellowship coordination, two oathbreakers are summoned as all enemies are crushed.
RGYBRG or GRBYGR - Thunder of the Oakenshield A truly flawless display of fellowship coordination, all fear and wounds are removed al all enemies are crushed.
YBRGYB or BYGRBY - Dawn on the Deep A truly flawless display of fellowship coordination, two oathbreakers are summoned as your target is heavily disabled.
YGRBYG or GYBRGY- Whispers under the Mountain A truly flawless display of fellowship coordination, all fear and wounds are removed as your target is heavily disabled.

These combinations don't appear to have their own name or effects:

RRRGGG - ? ?
RRRBBB - ? ?
BBBGGG - ? ?
BBBYYY - ? ?
BBBRRR - ? ?
GGGYYY - ? ?
GGGRRR - ? ?
GGGBBB - ? ?
YYYBBB - ? ?
YYYGGG - ? ?

YYYRRR - ? ?
YYYYYB - ? ?
YYYYYR - ? ?
YYYYYG - ? ?
BBBBBG - ? ?
BBBBBY - ? ?
BBBBBR - ? ?
GGGGGY - ? ?
GGGGGR - ? ?
GGGGGB - ? ?

After playing a burglar up to 38, fellowshipped with my kinship or with pick-up groups, I can offer the following advice for doing conjunctions:

If you have only 2 people, try for pairs (GG, or BB, or whatever).

If you have 3, try for flushes (in most cases) or simple straights (GYB is my favorite). Only try for the straight when you are using the same straight in every situation, so people get used to it. In this case you must assign an order so players know when to press. With a decent fellowship, this is not too hard with just 3.

With 4 or more, you may want to avoid straights. Although they work well, they hard enough that someone usually fails to contribute the correct color in time. Instead, assign your fellowship into 2 groups. Each group gets assigned a color. Usually the burglar will announce the colors before he opens: "Opening, BLUE - GREEN!" (this works best if everyone has voice-chat turned on).

When the wheel pops up, everyone in group 1 immediately chooses BLUE (for example). Once group 1 finishes (but not before), everyone in group 2 chooses GREEN. I've had the best results when the melee classes are in group 1. Put hunters (who have long induction timers) and minstrels (who have to switch targets from friendly to foe) in group 2.

If you have 4, try flushes or pairs (like GGBB); all combos work.

If you have 5, try for full houses (like GGGBB or YYYRR). Remember though with these that you can either do damage, or restore, not both -- so don't mix the colors.

If you have 6, try for the 5-1 combos (like RRRRRG). These work very well, and are not too difficult. We usually have our minstrel (if we have one) go last, since he has to switch targets to the mob. Everyone else just hits the assigned color when the wheel pops up, and the minstrel can decide the kicker color, based on what the group needs. Minstrels usually know exactly what the group needs.

With 6 you can also try for triples (like RRRYYY). These work pretty well, because if the last person doesn't join in time, you still get the full house. GGGBBB is very nice.

One last tip: under the Key Mapping Options, you can bind the four wheel options (up, down, left, and right) to keys -- I use the arrow keys. For me, it's a lot easier to hit the key, than to try and find my mouse cursor, and click on the right color.