How to use the words "will", "would", "can", "could", "shall" and "should"

The use of these modal auxiliaries is really too complex to answer simply, but I'll try to give a quick answer.

Notice in my sentence above "I'll try." The "will" expresses willingness, sometimes called volition. Sometimes it just expresses future time, but in current English "be going to" and simple present can also do this.

WOULD is often called the past of will, but it also expresses other thoughts. It can mean willingness to perform a hypothetical action, such as, "I would help you if you needed help." In this sentence, you do not need help, but I would help if you did.

WOULD can also be used to express habit in the past, as in "When I was a child, I would sleep with my teddy bear."

CAN expresses ability. Could is often called the past of can, and is used the way would is used in a hypothetical situation, where the speaker wants to express ability instead of willingness.

COULD sometimes also is used to make suggestions. Example: We could go to the beach, or we could go to the park.

SHALL is not much used in American English. Its use is mostly confined to utterances such as, “Let’s go, shall we?" "Shall I pick you up at ten?"

SHOULD expresses advisability--that something is a good idea. If I think you are sick, I say, "You should go to the doctor."

Less often we use SHOULD to make a prediction about something we think will happen. Example: The train should come in five minutes.

Source: http://in.answers.yahoo.com/question/index?qid=20080911090510AAZ5ji7