Draupadi's relationship with the Pandavas is complex and at the heart of the Mahabharat.
Draupadi and Arjuna
Let's jump right in with the most important relationship: Draupadi and Arjuna.
Of the five Pandavas, Draupadi favors Arjuna the most. She is in love with him, whereas the others are in love with her. Arjuna has won her in the Swayamvar, Arjuna is her Lord, and she is devoted to him. 1
On the other hand, she is not Arjuna's favorite wife. Arjuna does not like sharing her with 4 other men (conjecture on my part). Arjuna's favorite wife is Subadhra, Krishna's half-sister. He also dotes on Abhimanyu (his son with Subadhra) over and above his sons from Draupadi and Chitrangada (Here is a quick question: Can you name his sons from Draupadi and Chitrangada? Off the top of your head? Exactly). All of Draupadi's husbands married other women, but the only time Draupadi gets upset and distraught is when she learns of Arjuna's marriage to Subadhra. Subadhra has to go to Draupadi dressed as a maid, just to assure her that she (Subadhra) will always be beneath Draupadi in status. 2
Draupadi and Yudhisthir
Now let's see the reason why Draupadi's life is a shambles, why she is the most cursed woman of her time, and one of the most important reasons behind the Mahabharat war: Draupadi's marriage to Yudhisthir.
Here is something we need to understand first: Yudhisthir is a bastard. This is not to be held against him - all Mahabharat characters are gray - but people tend to forget this bit. Yudhisthir does not win Draupadi in the Swayamvar, he has no right to her. He lusts for her, he cannot bear seeing her everyday and not be able to have her. So he takes a small chance that fate throws his way, when Kunti says, "Share whatever you have between yourselves", and bullies Draupadi and his brothers into the weird "Lets all marry her" situation. Bhima does not like this, he claims that it is not right and that people will laugh at them. Yudhisthir tells him of Rishis who have done this before, and that it is accepted in Dharma.3 He then rushes forward and says that since he is the eldest, he must get "first dibs" with Draupadi. (Sorry to be crude here, but I tell it like I read it.) The brothers marry her according to age, eldest to youngest.
Then, Yudhisthir calls an assembly with his brothers and tells them the story of 2 powerful rakshasas, Sunda and Upasunda 4, whose love for the same woman led them to destroy each other. He says that the lesson to learn here is that the brothers must be careful when sharing Draupadi. She must be with one brother for a set period of time, and during this period the other brothers cannot touch her (carnally, that is). Yudhisthir decides that Draupadi will live for 1 year with each brother and that since he is the eldest, she will start the cycle with him.
Yudhisthir doesn't stop with this. He knows that Draupadi loves Arjuna. He does not want her to tempt him into breaking the rules. Hence he says that the brother who breaks this rule will have to go into exile for 12 years (seriously?). Further, the same punishment will apply if any brother happens to disturb another when he is carnally engaged with Draupadi. Ouch. This punishment actually comes into play when Arjuna disturbs Yudhisthir and Draupadi. Arjuna has to retrieve his weapons from the armory, in order to help a poor Brahmin whose cows have been stolen by thieves. Guess where Yudhisthir and Draupadi are doing it? That's right. The Armory.5
Arjuna departs on his 12 year tour of India, where he visits his father Indra, gets cursed by Urvashi, learns a lot of new skills from multiple teachers (Shiva, Indra etc), meets and marries Subadhra, followed by Chitrangada, etc. However, what happens to the year that he is to spend with Draupadi? It reverts back to Yudhisthir, who promises to care for Draupadi on Arjuna's behalf. Naturally.
Draupadi and Bhima
After that morose tale of manipulation, let's visit to a slightly more cheerful tale of manipulation: Draupadi and Bhima
Bhima is silly putty in Draupadi's hands. Of all her husbands, he is the one who loves her the most. He fulfills her every request, he cannot bear to see her hurt.
Who brings her flowers from Kuber's garden? Bhima. Who cries because his beautiful wife will have to serve as a Sairandhri (maid) to Queen Sudeshna of Matsya? Bhima. Who kills a 100 Kauravas to avenge the insult to Draupadi? Bhima. Who does Draupadi run to when she is molested by Keechak in Matsya kingdom? Bhima.
The other Pandavas are not under Draupadi's thumb. She is prone to outbursts of rage, she makes unreasonable, unwise demands. When she wants Keechak killed for molesting her, Yudhisthir tells her that it would expose their presence in Matsya kingdom, and advises her to "live with it". (Model husband that). Bhima simply walks up to Keechak in the middle of the night and tears him limb from limb. No questions asked.6
Draupadi shows us Bhima's human side. He is a savage monster with others, but he is always and only tender when it comes to Draupadi.
Draupadi with Nakul and Sahadev
As with most of the Mahabharat, Nakul and Sahadev don't really matter here. I haven't read any version of the Mahabharat where Nakul and Sahadev have any role of substance. In reality, Nakul and Sahadev are more loyal to Yudhisthir than anyone else. They don't share father or mother with Yudhisthir, yet they follow him everywhere and do exactly as he asks. They could have gone and ruled over Madradesh (their birthright through Madri), and lived a life of luxury and ease, but they stuck with their brother through thick and thin. Makes one appreciate them a little bit more.
In summary, Draupadi's curse is the curse of beauty. She is the object of every man's lust, but no one cares much for what she desires or feels. Her husbands gamble her away as if she were property. When Dusshasana strips her in view of a full court, she has to beg Krishna to save her. Her husbands don't lift a finger. Even at the end of their 13 year exile, the Pandavas are not intent on war. They worry that the losses in the Kurukshetra war will be too big to warrant it. Draupadi has to turn to her friend, Krishna, to heal her soul. Krishna promises her:7
"Soon wilt thou, O Draupadi, behold the ladies of Bharata's race weep as thou dost. Even they, O timid one, will weep like thee, their kinsmen and friends being slain. They with whom, O lady, thou art angry, have their kinsmen and warriors already slain.... I will accomplish all this."
And thus comes about the Mahabharat war.