1. The conditional
Dr. Silvestri

2. Uses same base as future simple regulars and irregulars
Uses same base as future simple regulars and irregulars. For regular verbs is entire infinitive.
The conditional endings are -ais, -ais, -ait, -ions, -iez, -aient (These are also the imperfect endings).

3. Uses
The French conditional is mainly used in if...then constructs. It expresses the idea that if this were to happen, then that would be the result. 
While French uses the word si in the "if" or condition clause, it does not use a term for "then" in the result clause. The conditional verb itself is used in the result (then) clause, while only four other tenses are permitted in the si clause: présent, passé composé, imparfait, and plus-que-parfait.
Il mangerait s'il avait faim. - He would eat if he were hungry.
Si nous étudiions, nous serions plus intelligents. - If we studied, (then) we would be smarter.
Il mangerait avec nous si nous l'invitions. - He would eat with us if we invited him.

4. Special notes for certain verbs
The verb vouloir (to want) is used in the conditional to express a polite request:
Je voudrais une pomme. - I would like an apple.
Je voudrais y aller avec vous. - I would like to go with you.
However, you can't say "si vous voudriez" to mean "if you would like," because the French conditional can never be used after si. You would say “si vous voulez”
The verb aimer (to like, love) is used to express a polite desire, sometimes one that cannot be fulfilled:
J'aimerais bien le voir ! - I would really like to see it!
J'aimerais y aller, mais je dois travailler. - I would like to go, but I have to work.