The first step is in understanding modal auxiliaries is to distinguish modal auxiliary verbs from auxiliary verbs. There are three auxiliary verbs. These are do, be, and have. As described by John Fleming of DeAnza College, auxiliary verbs are used in specific instances and may also uesd as main verbs.
Auxiliary have is used to construct the perfect aspect in the three tenses (past, present, future): Perfect aspect have + -ed participle. Auxiliary have is used in all moods and in all affirmative and negated sentences.
Auxiliary be is used in continuous aspects (Progressive and Perfect Progressive): Progressive aspect be + -ing participle; Perfect Progressive have + be + -ing participle. Auxiliary be is a significant part of passive voice sentence construction: be + -ed participle. You'll note that the difference in construction between the progressive aspect and the passive voice is the form of the participle: progressive aspect uses the present -ing participle while passive voice uses the past -ed participle. Auxiliary be is used in all moods and in all affirmative and negated sentences.
Auxiliary do is used in simple past tense and simple present tense. Auxiliary do differs from the other two auxiliary verbs because it is used only in interrogative mood and in negated sentences.
As taught by Howard Jackson of Birmingham City University, and others, modal auxiliaries, in contrast to auxiliary verbs, are of greater variety. The modal auxiliaries are can, could, must, may, might, will, would, ought to, shall, should. Some people add used to, need, dare, but the addition is not necessarily common. Modal auxiliaries fulfil a specialized function in English. They express futurity and probability along with obligation and politeness. Some people give a more expanded explanation and elaborate on the above three categories by saying modal auxiliaries (modals) express advice, ability, necessity, expectation, permission, possibility and more, but all of these are subcategories of futurity and probability and obligation and politeness. Modals establish relationships between individuals in written or spoken discourse and establish the distinctions between obligation and discretionary choice.
English in fact has no inflected future tense. In English, future tense is a construct of will, shall, would, or should with a main verb: "I will be there." "I shall come to you." "You should work harder to graduate." "I would run more if I had more time." Therefore modals are integral to expressing futurity in English.
Probability, the degrees of possibility, impossibility, and certainty are expressed with must, may, can, and might. "The invitation must have been sent." "It may be lost." "Invitations can be misdirected." "It might have been misdirected."
Obligation and politeness are expressed variably through must, ought to, may, will, could, shall, might (formal), would, should, can, and sometimes need: "Will/would/could/can you help?" "May/can/might I enquire your name?" "You must/need to/ought to/should help her."