The modal verb must is used with a bare infinitive to express the idea that something is necessary or obligatory or that something is certain or probable. Need to and have to also express the idea of being necessary or obligatory. The past tense of must is must, but it is used only in reported speech or in certain senses of the word. Had to is often used as the past tense of must. There are more details in the sections below.
The main use of the modal verb will is to form the future tense of lexical verbs, but it has other uses as well, such as asking someone to do something or saying that you are willing to do something. Shall is sometimes used to form future tenses with I and we, but has other uses, such as asking for advice or expressing intentions. Would and should are the past tenses of will and shall, but they have a number of other uses.
The modal verb may is used with a bare infinitive to express the idea that something is possible or permitted. It is also used to express opinions and wishes. Might is the past tense of may, but has a number of other meanings. May is used to ask for, give or refuse permission. If you want to be a little more polite or formal in making a request, use might. In reported speech, might is rather formal and a little old-fashioned. Could would be more likely in everyday speech.
The modal verb can is used with a bare infinitive to express the idea that something is possible or that someone is able to do something or allowed to do something. Could is the past tense of can, but is also used in a number of other ways. Can and could are often used with verbs that describe the use of the mind or the senses (sight, hearing, touch, etc), but without much idea of ‘possibility’ or ‘ability’ at all. Could is also used to ask permission. It is slightly more polite than can. Can may be used to describe something that sometimes or frequently happens.
Modal verbs, also known as modal auxiliaries or medals, are mainly used to add to the lexical verb a feeling of the action being, for example, possible, likely, necessary, certain, compulsory, allowed or advisable.