Describe the conspiracy among Romeo, Juliet, the Nurse, and Friar Laurence in Romeo and Juliet, and what is the result?
Romeo and Juliet are secretly married, between Acts 2 and 3, and the Nurse and Friar Laurence, who performs the rite, are the only ones who know about it.
Friar Laurence's ultimate goal, in marrying Romeo and Juliet, is to end the feud between the Montagues and the Capulets:
In one respect I'll thy assistant be;
For this alliance may so happy prove
To turn your households' rancor to pure love.
He recognizes Romeo's sudden change of heart (he was miserable because of his unrequited love for Rosaline, then forgot about her as soon as he saw Juliet), and cautions him to "love moderately," advising, "Wisely and slow. They stumble that run fast." In a haunting example of foreshadowing, just before the marriage ceremony, Friar Laurence says,
These violent delights have violent ends
And in their triumph die, like fire and powder
which as they kiss consume.
Juliet's Nurse, who is the only other character aware of the relationship between Romeo and Juliet (she was a messenger between Romeo and Juliet, and essentially arranged for Juliet to go to church to be married), loses Juliet's confidence once Capulet announces that Juliet will marry Paris:
I think it bese you married with the County.
O, he's a lovely gentleman!
Romeo's a dishclout to him.
Though the Nurse has good intentions in counseling Juliet (especially since Romeo has been banished for killing Tybalt, and presumably Juliet won't be able to see him anymore), Juliet sees her stance as a betrayal. From this point, Juliet does not include her nurse in any plans that concern Romeo (specifically, her plan to take a sleeping potion that will make her appear dead).
The result of this "conspiracy" is obviously the deaths of Romeo and Juliet. At the same time, though, the Capulets and Montague have vowed to end the ancient feud they've been engaged in for so many years.
There are a couple of events enacted by Romeo and Juliet behind the backs of their families in the play, but the only one in which all the characters you mention are complicit is the wedding.
In the balcony scene (Act II, Scene ii), Juliet asks Romeo if he will, in essence, put his money where his mouth is and marry her. If so, he should find out when and where. Romeo confides his wish to marry Juliet to the Friar, who agrees to marry them, hoping that this will put an end to the Capulet/Montague feud. Romeo then tells the Nurse (who has been sent by Juliet to find out from him about the wedding plans) to have Juliet meet him at the Friar's cell. And so they are married, with the help of the Nurse and Friar.
The result of this secret marriage is that Juliet must later pretend to die (with the potion given her by the Friar) in order to avoid bigamy by marrying Paris. And of course it is the false report of Juliet's death to Romeo that causes him to return to her side to take his own life. Which causes her to take her own, once she wakes and finds him dead. Which causes the ending of the feud. So, ultimately, the Friar gets his desire, only not necessarily by the means he might have wished.