You are allowed to pick multible Daedras if your not sure about which to chose or if there is some other unknown reason why you want to chose more than one.
So, the normal gamer may perhaps not know who all those guys are so I will give a brief explanation of each Daedra Prince (copy-pasted from http://http://www.uesp.net/wiki/Lore:Daedric_Princes, since I am a lazy bastard):
Azura.
Moonshadow is Azura's plane of Oblivion, where she lives in a rose palace. According to The Doors of Oblivion, it is blindingly beautiful and colorful, with flowers, waterfalls, trees, and a city of silver. The Winged Twilights are her servants.
Azura was the god-ancestor that taught the Chimer the mysteries needed to be different from the Altmer. Some of her more conventional teachings are sometimes attributed to Boethiah. In the stories, Azura is often more a communal cosmic force for the race as a whole than an ancestor or a god.
Own input: In Elsweyr Azura (the Khajiits call her Azurah) is credited for making the Khajiit out of the Bosmer stock in Elsweyr.
Azura is mostly depicted as a female and when she shows herself for a mortal she is in a female form.
Azura cursed the Chimer after they betrayed Nerevar Indoril and made their skin grey and their eyes red.
Some people says that she acts like a spoiled princess and that she is extremly petty.
Boethiah.
Heralded by the Prophet Veloth, Boethiah is the original god-ancestor of the Dark Elves. Through his illuminations, the eventual Chimer, or Changed Folk, renounced all ties to the Aldmer and founded a new nation based on Daedric principles. All manner of Dark Elven cultural 'advances' are attributed to Boethiah, from philosophy to magic to 'responsible' architecture. Ancient Velothi allegories are uniformly heroic successes of Boethiah over enemies of every type, foundation stories of Chimeri struggle.
Own input: It is rumoured that Boethiah helped Jagar Tharn during his betrayal.
Goldbrand and the Ebony Mail are his most common artefacts.
(EDIT nr. 2)Boethiah also ate the Aedra Trinimac and popped out Malacath when a group of aldmer loyal to Trinimac tried to stop the chimer exodus.
Clavicus Vile.
Own input: It is rumoured that Clavicus helped Jagar Tharn during his betrayal.
Clavicus Vile granted the Vampires the ability to blend into mortal society undetected.
The Bittercup and the Mask of Clavicus Vile are his most common artefacts.
Clavicus likes souls for the sake of having them.
Hermaeus Mora.
Apocrypha is Hermaeus Mora's plane of Oblivion, an endless library where all forbidden knowledge can be found. The books all have black covers with no titles, and the library is haunted by ghosts forever searching for knowledge. Apocrypha is Greek for hidden-things.
Own input: Mora nearly seduced the Nords into becoming Aldmer.
It seems that Herma-Mora is depicted as some evil thing who takes villagers lost in the forest in Nedic myth...
The Oghma Infinium is his (but the sage Xarxes wrote it) most common artefact.
Hircine.
Own input: His artefacts is Hircines Ring and Savior′s Hide.
Jyggalag
Shivering Isles reveals that Jyggalag is in fact Sheogorath, a fellow Daedric Prince. Jyggalag had grown in power, and the other Daedric princes had grown fearful and jealous of him. They cursed him to live as Sheogorath, the incarnation of the thing he hated most, madness. This is also why he does not show up in the Elder Scrolls universe apart from TSI, he spends most of the time as Sheogorath. He was, however, allowed to return to his true form at the end of every era in order to retake the Shivering Isles from his mad alter-ego in an event known as the Greymarch. However, once this was done, Jyggalag would be transformed back into Sheogorath, starting the cycle again. At the end of the third era, at the dawn of the Greymarch, a lone Champion of Sheogorath was given the chance to defeat Jyggalag, halting the Greymarch before it could conquer Sheogorath's Realm. This released Jyggalag from his curse, allowing him to remain in his true form. The Champion was then permitted to assume Sheogorath's station in the Shivering Isles. It is unknown yet if Jyggalag will ever return to the Shivering Isles in an attempt to retake them, or what will happen to the Champion of Cyrodiil as a result of their new position as the Lord of Madness.
Own input: He is pretty unknown. Do you like that everything looks like crystals and no-one got any creativity? Propably not. Well that is how Jyggalag wants it to be.
Malacath.
Malacath's plane of Oblivion is Ashpit. It is described as consisting only of dust, palaces of smoke, and vaporous creatures: "anguish, betrayal, and broken promises like ash filled the bitter air". Levitation and magical breathing are necessary to survive. The hideous Ogrim are his servants.
Own input: His most common artefact is Volendrung.
He seems to like the Orcs...
Boethiah also ate the Aedra Trinimac and popped out Malacath when a group of aldmer loyal to Trinimac tried to stop the chimer exodus.
Mehrunes Dagon.
Own input: He definitely helped Jagar Tharn during his betrayal.
He invaded and conquered the Battlespire to assist Jagar Tharn.
He destroyed the city of Mournhold and Ald Sotha (Home of House Sotha and Sotha Sil′s birthplace) at the end of the first era.
His most common artefacts are Mehrunes Razor and the Daedric Crescents (only one is still left in the world of Tamriel).
He tries to invade and conquer Cyrodiil at the last year of the third era.
Mephala.
Own input: Her/His artefacts are the Ring of Khajiit and the Ebony Blade.
Meridia.
The Hero of Daggerfall received The Ring of Khajiiti from her after defeating a sorcerer who betrayed her. Her enemies are Ebonarm and Molag Bal. Her day of summoning is the 13th of Morningstar.
Meridia's strong dislike of undead beings is shown in the quest that the player can do for her in Oblivion. Meridia presents the player with the task of destroying Necromancers who are raiding tombs for corpses to raise an army of the undead. Meridia has a great and everlasting hatred for the undead and will greatly reward any who eliminate them from the world.
Own input: There is very little information abot her.
Molag Bal.
Molag Bal's plane of Oblivion is Coldharbour. The book The Doors of Oblivion says that his plane resembles a copy of Nirn, including the Imperial Palace, but all desecrated and ruined. The ground is sludge, the sky is on fire, and the air is freezing. The Daedroths are typically counted among his servants.
Own input: He is said to be the father (along with VIVEC!! Whom he seduced) of a population of mutant degenerates living in the Molag Amur region in Vvardenfell!
He may have assisted Jagar Tharn during his betrayal.
He was responsible for destroying the Bosmeri town of Gilverglade during the end of the first era.
The Mace of Molag Bal is his most common artefact.
He likes havoc and other such crazy stuff.
Namira.
Own input: Namira′s Ring is her most common artefact.
Nocturnal.
Nocturnal rewarded The Hero of Daggerfall with the The Skeleton's Key in exchange for slaying a rogue mage who caused her much grief. Her enemies include Ebonarm and Azura. She has no known allies. She may be summoned on Hearthfire 8th.
Nocturnal rewarded The Champion of Cyrodiil with the Skeleton Key in exchange for recovering an artifact known as the Eye of Nocturnal which had been stolen by a pair of Argonians, Bejeen and Weebam-Na.
Own input: The skeleton key is her most common artefact.
Peryite.
Own input: His most common artefact is the Spell Breaker.
Sanguine.
The term sanguine can mean either "cheerfully optimistic" or "bloody"; the double meaning is appropriate for a prince whose realm encompasses both the light and dark sides of passion.
Own input: "He is a merry good ol′ fellow..."
Sheogorath.
Own input: Sheogorath′s most common artefacts are Wabbajack, the Fork of Horripilation, Gambolpuddy and the Staff of the Everscamp.
In Elsweyr Sheogorath is known as Sheggorath or "The Skooma Cat".
It was he who threw the Ministry of Truth on Vivec City (although it was stopped by Vivec himself... for a while).
Most think that Sheogorath is pretty funny.
He is clad as a harmless banker to lead people down the road to madness easier.
Madness is funny
Vaermina.
Vaermina's plane of Oblivion is Quagmire. It is described in The Doors of Oblivion as a nightmare realm, where every few minutes reality shifts and becomes ever more horrifying: "a dark castle one moment, a den of ravening beasts the next, a moonlit swamp, a coffin where he was buried alive."
The Hero of Daggerfall received The Skull of Corruption from Vaernima. She is an enemy of Ebonarm, Boethiah, Peryite, and Hermaeus Mora, but is allied with Sanguine. Her summoning day is the 10th of Sun's Height.
In Morrowind, Vaermina is referenced at the end of the Cure for Vampirism quest in which Molag Bal obtains the cure from her to present to the Nerevarine.
In Oblivion, the player could receive the Skull of Corruption from Vaermina by venturing into the tower that manifests the nightmares experienced by a man who stole Vaermina's orb.
Own input: She likes to send nightmares to people! Scary stuff...
Her most common artefact is the Skull of Corruption.
I really hope that the information is helpful since I have used lots on time for it!
Now vote!
Lorkhan is not included since we got no confirmed source that (EDIT nr.1) he′s a Daedric Prince.