The first word of each pair is the present tense and the second, the past tense. However, these days, they have some differences in usage. I'll try to give you the main uses. Shall is now used to make offers or suggestion in the form of a question, e.g. 'Shall I call you later?' / 'Shall we stay home tonight and watch a film?' Should is primarily used to offer advice or to say something is a goo idea, e.g. 'Everyone should eat plenty of fruit and vegetables.' May / Might are used to say that something is possible in the future, e.g. 'It may/might rain later'. They can also be used to ask permission but 'may' is more common for this, e.g. 'May I have a glass of water?' Can / could are primarily for ability. e.g.s 'I could read before I started school'. 'I can ride a bike'. They can also be used to ask permission, e.g. 'Can I use your bathroom?' 'Could I borrow your car?' 'Could' is more formal here. 'Will' is, of course, used to indicate the future, and in real conditional sentences, e.g. 'If my boss gives me a pay rise, I will be able to buy a new car'. 'Would' is used in unreal conditional sentences, e.g. 'If I were rich, I would be happier.' It is also used to make polite requests, e.g. 'Would you lend me £10?' 'I would like a sandwich and a Coke.'

"Shall" is like like "will" but more formal and old-sounding. It implies certainty, whereas "should" is a suggestion. i.e. I shall go to the beach (mostly likely WILL go if not for sure) vs. I should go to the beach (maybe?) "May" and "might" are basically the same. i.e. I may walk the dogs later vs. I might walk the dogs later. "Can" and "could" are the same verb but in difference tenses. "Can" is present tense and says you have the ability to do something with certainty. "Could" is conditional tense and implies doubt or uncertainty as to what you "could" do. i.e. I can go to the concert (you definitely can go) vs. I could go to the concert (but do you want to?) "Will" is the same as shall but more commonly used. "Would" is conditional auxiliary and goes with any verb to add a variety of differences. i.e. Would you like to get some ice cream? (very passive) vs. He would want you to have his dog. (Implication) --It's very arbitrary. English is convoluted >.>

This is a very, very big question. :)