Which is the correct use of these two words, and in which context should one be used rather than the other?
Here is a good description of when to use shall:
...shall is used for the future tense with the first-person pronouns I and We: I shall, we shall. Will is used with the first-person (again, I refer to traditional usage) only when we wish to express determination. The opposite is true for the second-person (you) and third-person (he, she, it, they) pronouns: Will is used in the future tense, and shall is used only when we wish to express determination or to emphasize certainty.
However, growing up as a native American speaker in Colorado, I never used "shall" in normal speech. However, I believe in questions it has become more common, e.g. Shall we go? but an American native speaker saying e.g. "I shall do that for you" will almost always sound affected or connote a Shakespearean context. I often have wondered in which parts of the world (England?) and which social echelons using "shall" in statements is actually still practiced by native speakers.
Don't forget that 'shall' at the start of a question is used to make a suggestion:
Shall we play tennis?
But 'will' at the start of a question does not have the same meaning:
Will we play tennis?
The note about the usage of shall in the New Oxford American Dictionary is the following:
There is considerable confusion about when to use shall and will.
The traditional rule in standard English is that shall is used with first person pronouns (I and we) to form the future tense, while will is used with second and third persons (you, he, she, it, they): I shall be late; she will not be there.
When expressing a strong determination to do something, the traditional rule is that will is used with the first person, and shall with the second and third persons: I will not tolerate this; you shall go to school.
In practice, however, shall and will are today used more or less interchangeably in statements (although not in questions). Given that the forms are frequently contracted (we'll, she'll, etc.), there is often no need to make a choice between shall and will, another factor no doubt instrumental in weakening the distinction. In modern English, the interchangeable use of shall and will is an acceptable part of standard U.S. and British English.
When I started my classes on British English, I was taught that I shall go home was the future tense of I go home; asking to a person living in the east coast of the USA, I learned that I shall be late (or I shall return) has a slightly different meaning from I will be late (or I will return), at least in some contexts.
"Shall" originally indicated owing to do something. "Will" originally, and sometimes still, indicates a desire to do something. Eventually, they both came to indicate the future, but "will" usurped the place of "shall," at least in my part of Virginia. As far as I can tell, "'ll" has since taken the place of will. :)
EDIT: Should have included a link or two to begin with, not to mention double-checked my answer. Whoops, sorry.
Etymology of "will": http://www.etymonline.com/index.php?term=will
And as opposed to "shall": http://www.etymonline.com/index.php?term=shall
Shall and will are both modal verbs in English used to express propositions about the future. According to the New Oxford Dictionary of English, In modern English the interchangeable use of shall and will is an acceptable part of standard British and US English.
Collins English Dictionary defines 'will' and 'shall' as follow:
Will: (you, he, she, it, they, or a noun as subject) used as an auxiliary to make the future tense.
Shall: (I or we as subject) used as an auxiliary to make the future tense.
I shall be at home tonight.
Will you come round.
Let us use 'will' instead of 'shall', if we want to emphasize. So:
I will be ready on time, I promise.
I do not care what your wicked stepmother says: you shall go to the ball.
But if you want to see what may happen when we reverse 'will' with 'shall', and vice versa, read the following sentences:
I shall drown, no one will save me. (I should like to be saved but everyone is ignoring me.)
I will drown, no one shall save me. (I am determined to drown. I forbid anyone to throw a life belt.)
In some situations will and shall may point to different sources of motivation to act in a particular way—namely, an internal source versus an external source. As an example (in the third-person singular) of the compulsory aspect of shall versus the volitional aspect of will, consider this exchange from Henry James's short story "Covering End":
“And what has she [Cora Prodmore, Mr. Prodmore's daughter],” he [Mr. Prodmore] appealed, “expected me to give up? What but the desire of my heart and the dream of my life? Captain Yule announced to me but a few minutes since his intention to offer her his hand.”
She [Mrs. Gracedew] faced him on it as over the table. “Well, if he does, I think he’ll simply find—”
“Find what?” They looked at each other hard.
“Why, that she won’t have it.”
Oh, Mr. Prodmore now sprang up. “She will!”
“She won’t!” Mrs. Gracedew more distinctly repeated.
“She shall!” returned her adversary, making for the staircase with the evident sense of where reinforcement might be most required.
Mrs. Gracedew, however, with a spring, was well before him. “She shan’t!” She spoke with positive passion and practically so barred the way that he stood arrested and bewildered, and they faced each other, for a flash, like enemies.
The progression here is from Mr. Prodmore's emphatic "She will!"—his assertion that Cora Prodmore will marry Captain Yule of her own volition—to his even more emphatic "She shall!"—his assertion that she will do so under pressure from Mr. Prodmore. Though in the first instance Mr. Prodmore has no doubt that his daughter accepts the inevitability of the marriage that he intends to bring about, his authoritarian intensity increases in the second instance as he shifts from expressing confidence in her intentions to expressing confidence in his ability to force his expectations upon her.
In some contexts, such as legal contracts or formal standards, shall indicates an obligation or command, whereas will might be interpreted as an observation or prediction.
shall/should is stronger and heavier in emphasis than will/would. Eg. statements that I can think off right off my mind for comparison in connotation would be:
He shall/should be doing it.
He will/would be doing it.
Another distinction I've heard people make is that "shall" implies a deliberate action and "will" implies a result ("I shall sing and the crystal will crack"). Strictly conversations, so no authorities to quote here.