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.
The modal verb can is used 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.
The modal verb may is used 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 and uses as well.
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 land 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 too have a number of other uses.
The modal verb must is used 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.