haber – to be (impersonal), to have, to have-to

he, has, ha, hemos, habéis, han

hay is almost a separate verb, only uses third person singular:

hay un coche – there’s a car

Also (auxiliary):

haber + past participle – to have (in compound verb forms)

(see below)

haber de + infinitive – to have to, must, should

has de pedirlo – you have to ask for it

han de venir hoy – they must come today

haber que + infinitive – must, have to (see also tener que)

hay que decírselo – we have to tell her

hay que estudiar mucho – one must study a lot .

hay que reservar la plaza de garaje? – is it necessary to (must one) reserve a parking space?

The regular Future tense was originally formed from infinitive + suffix from haber present tense (he, has, ha, hemos, h[ab]éis, han); e.g., comer-he, comer-has, etc.
The Conditional was similarly formed from infinitive + suffix from haber imperfect tense (había, habías, había, habíamos, habíais, habían); e.g., comer-ía, comer-ías, etc.

Conjugation

Present

he, has, ha, hemos, habéis, han

→ used with past participle of a verb to form perfect, e.g.:

He perdido mí pasaporte – I have lost my passport

¿Qué ha pasado? – What has happened?

Preterite

hube, hubiste, hubo, hubimos, hubisteis, hubieron

→ used (literary) with past participle of a verb to form past anterior, e.g.:

Miguel hubo estado en la casa – Miguel had been in the house

Imperfect

había, habías, había, habíamos, habíais, habían

→ used with past participle of a verb to form pluperfect (past perfect), e.g.:

Yo siempre había querido ser ingeniero – I had always wanted to be an engineer

había cuatro libros – there were four books

Future

habré, habrás, habrá, habremos, habréis, habrán

→ used with past participle of a verb to form future perfect, e.g.:

Habré terminado mi report para mayo. – I will have finished my report by May.

Subjunctive

haya, hayas, haya, hayamos, hayáis, hayan