Shakespeare is using a metaphor here wherein Romeo compares his meeting and falling in love with Juliet to a ledger, or a record of debts and payments. As my colleague stated, Romeo has just learned Juliet's identity. She is a Capulet, his families sworn enemy. So, when he says, "Oh dear account! My life is my foe's debt," what he means is that he is forever in debt to his enemy for bringing him his love, Juliet. The account is dear in two ways. Juliet is dear to him. Also, the debt is dear, meaning very costly; he fears his enemy Capulet will not accept his "payment" of love and gratitude--that Capulet will keep him from his dear Juliet.
Enotes has wonderful side-by-side texts for Shakespeare's plays. They have the original text and a modern translation. Check out the link below for the e-text for Romeo and Juliet.
This is from Act I, Scene 5 when Romeo has just been told that Juliet is a Capulet. He saw her at the party and fell in love with her before he knew her name. He already feels he can't live without her -- "my life" --and realizes he is supposed to hate her since he is a Montague, and she is a Capulet.
This quote takes place at the end of Act 1, scene 5. Romeo and Juliet have just fallen in love, essentially at first sight. Taken with each other, they haven't even exchanged names, choosing instead to flirt and share a kiss at the Capulet party. When their romantic moment is cut short by the Nurse, Juliet heeds her mother's call and leaves. Romeo takes this opportunity to find out more about the woman he has fallen in love with. When he asks after the identity of Juliet's mother, the Nurse tells him that she is the "lady of the house" and that any man lucky enough to marry Juliet "...shall have the chinks." (1.5.113-117)
At this news, Romeo's heart sinks. He says in an aside, "Is she a Capulet? O dear account! My life is my foe’s debt." (1.5.118) Realizing that she is a direct descendent of his own family's mortal enemy, he makes mention of the price of love in his use of the word "account". This indicates that his life is no longer his own, but rather, Juliet's. In other words, Romeo's enemy has control of his life from this point on.
Romeo says this in Act I, Scene 5. What he is referring to is the fact that he has just found out that Juliet is a Capulet. It is too late for him -- he is already totally in love with her. So that is why he says this.
What he means is that his life now belongs to his foe. This is because he is so in love with Juliet that she is his whole life. And since she "belongs" to Lord Capulet, Lord Capulet (who is the enemy of Romeo's family) holds Romeo's life in his hands.
The "dear account" part means it's a big price (dear means expensive here) to pay. The Nurse has just told him that whoever marries Juliet gets a lot of money, Romeo says he would be paying a high price for it.