Modal auxiliaries generally express speakers attitudes. Modals can express whether a speaker feels something is:

• necessary (must, need to, have to)
• advisable (should, ought to, had better)
• permissible (may, can, could, might, would you mind)
• possible (may, might, could)
• probable

And a modal can express the strength of the attitude. Each modal has more than one meaning. 

Modals do not take -s  (agreement), and they are followed by the simple verb form (exc. ought).    (Azar 9-1, 10-1)

Nine central modal auxiliaries: can, could, may, might, must, shall, should, will, would  — They differ from other verbs both lexical verbs and primary auxiliaries, in that they have no nonfinite forms (but have contracted forms I'd, can't ).

Modals and semi-modals can be grouped into three categories: (485)

• permission/ possibility / ability: can, could, may, might
• obligation / necessity: must, should, had better, have (got) to, need to ought to, be supposed to
• volition / prediction: will, would, shall, be going to

Modal auxiliaries:

• are invariant forms
• precede the negative particle "not"
• precede the subject in yes-no questions
• are followed by bare infinitive (without "to")
• cannot co-occur with each other (*might could)

Marginal auxiliaries → dare (to), used (to).
Semi-modals → ought (to), have (to), need (to), be going (to), had better, have got (to)

The modals — can, could, may, might, will would, shall, should, must and ought (Br-Eng) — are called modal auxiliary verbs. They:

• have no -s in third person singular
• have questions, negatives, and tags made without do
• are followed by modal infinitive without "to".
• have no infinitive or past forms (exc. would, should, could, might in subordinated clauses)
• express past ideas with a perfect form (have + participle) should have been, should have known, etc.

"We use them [modals], for example, to talk about things we expect, which are or are not possible, which we think are necessary, which we want to happen, which we are not sure about, which tend to happen, or which have not happened." (Swan 353)

Modal Auxiliaries have five distinct properties:

• primary forms only (no inflectional forms)
• no person or number  agreement
• bare infinitival complement (plain form – no "to"
• a required verb form in a "remote conditional" (hypothetical)
• modally remote preterit in a main clause — could, might, would, should. (Could / would you help me please?

See "Modality" below.

Each modal can have two types of meaning:

• intrinsic → actions and events that humans (or other agents) control directly: permission, obligation, volition (or intention)
• extrinsic → logical status of events or states; assessments of likelihood: possibility, necessity, or prediction

(Biber  6.6.0-6.6.4)

Modality    (3 §9.2.2)

• epistemic → what the speaker believes is probable; likelihood based on inference (factuality)  He must / may / might be here.
• deontic → what the speaker feels is right morally or feels is right for the sake of expediency (actualization)  He must / should / could do the work.
• dynamic → the subject's own ability or willingness to act. (Dynamic modality does not express the speaker's opinion, nor does the speaker affect the situation.)   He can speak Swahili. (ability) I will have lunch. (volition)

• Strong vs. Weak Modality → the strength of the speaker's commitment to the factuality or actualization.  → He must be here.  vs. He may be here. (See external vs. internal negation (Huddleston 3 §9.2.1)
• Medium Strength Modality → He should be here.