The use of 'shall' is declining especially in American English, but traditionally 'shall' is used for the future tense with the first person (singular and plural) in British English instead of will. I shall wait = I will wait. We shall be late = We will be late. (Today this is considered quite formal but it is still used) Shall is also used to express determination or compulsion to do something with the second and third person instead of will. 'You shall go to the party if you want to!' To express determination or compulsion with the first person we use will. 'I will go to the party if I want to.' Shall is also used to make questions which are offers/suggestions/ requests for permission in the first person instead of will. 'Shall we dance?' Shall I make you a cup of tea?' 'Shall I call the office?' Shall is also used in legal documents/ contracts to express mandatory action and prohibition 'The lessor shall not be held liable for any damages due to natural calamity or acts of God..'

I agree with the above; that is the most correct definition, but you might also encounter "shall" in question form as in, "what shall we do?" In this case, it can also be understood as "should."

The previous answer is correct. Shall can replace will, but shall is less common and dated. In speech and informal e-mail use 'll. I'll, we'll, etc. In letters will is more common than shall.

"Shall" express something will happen for certain, this is an auxiliary verb. I shall see you tomorrow for dinner. (The speaker is certain that he/she will show up for dinner)