How do you construct the past tense in French?

Unlike in the English past tense, the French have two auxiliary 'helping' verbs - avoir and être, these become before the past participle, e.g. descendu. For example, j'ai mangé, and je suis allé.
So as you can see here, the structure is 'who has done it (subject) + helping verb (auxiliary) + second verb (past participle)'
These auxiliaries are formed as follows
subject   avoir    être   
je       ai      suis
tu       as     es
il/elle/on  a      est
nous     avons   sommes
vous     avez    êtes
ils/elles   ont      sont
Which one of these you use depends on the verb that follows/the past participle, for example a minority of 'moving' verbs mean that you must use être, e.g. je suis... These can be remembered using the anacronym DR & MRS VANDERTRAMP, they are:
Devenir - to become
Revenir - to come back
&
Monter - to climb
Rester - to stay
Sortir - to go out
Venir - to come
Aller - to go
Naître - to be born
Descendre - to descend
Entrer - to enter
Rentrer - to return
Tomber - to fall
Retourner - to return
Arriver - to arrive
Mourir - to die
Partir - to leave
All other verbs require the use of avoir as the helping verb/auxiliary.
The second verb/past participle, e.g. venir, manger, etc, are formed from the infinitive (as you find them in the dictionary) as follows:
Verbs ending in -er (e.g. manger) lose the -er and add an -é; e.g. mangé
Verbs ending in -ir (e.g. partir) lose the -ir and add an -i; e.g. parti
Verbs ending in -re (e.g. rendre) lose the -re and add an -u; e.g. rendu
verbs ending in -oir (e.g. voir) lose the -oir and add an -u; e.g. vu
The final step to forming the past is that only DR & MRS VANDERTRAMP verbs must take an agreement - additional letter(s) - of gender and quantity depending on who has done the verb. For example
gender     singular     plural
masculine  - (no addition) - s
feminine   - e         - es
e.g. elle est allée (added e because subject is feminine); Marie et Claire sont descendues (added es because subjects are both feminine; Marc et Tom ont devenus (just the addition of an s because whilst there are two subjects, neither are feminine)