French conjugations come from Latin, with more or less differences (it's more direct in spanish by example) depending on tenses and verb group.
Have a look at "amare" here, and compare with French "aimer" there.
- Amo ====> j'aime
- Amas ===> tu aimes
- Amat ===> il aime (lost the ending t a long time ago)
- Amamus => nous aimons (something like amus => aums => oms => ons)
- Amatis => vous aimez (something like atis => aits => aiz => ez)
- Amant => ils aiment
NB: Latin short "a" often became "e" in French...
Formation with to be and to have
In latin, lots of tenses are constructed with the addition of "to be".
- Future perfect is basically perfect radical + future of to be (amav + erit => amaverit)
- Pluperfect is perfect radical + imperfect of to be (amaverat)
Imperfect 2nd conjugation
About "je finissais", the imperfect is always formed by adding "ais, ais, ait, ions, iez, aient" to the 1st person plural radical of the present form, so "nous FINISSons" ==> "je finissais, etc.". The "s" is doubled to avoid sounding like a "z", it is not an irregularity (or if you prefer, the writing is irregular so the pronunciation isn't).
This said, you should have a look at finire. Pluperfect subjonctive looked like this : "finivíssem finivísses finivísset finivissémus finivissétis finivíssent", and in French "que je finisse, que tu finisses, qu'il finisse, que nous finissions, que vous finissiez, qu'ils finissent". Very similar, is it not?
The praeterit comes directly from Latin as well. Latin have more conjugations than French (5 vs 3), but to simplify it's like Latin's first conjugation (are) became French's first (er), Latin's "ire" verbs became French's second, and the reste French's third.
- Amavi ====> j'aimai
- Amavisti ===> tu aimas
- Amavit ===> il aima (lost the ending t a long time ago too)
- Amavimus => nous aimâmes
- Amavistis => vous aimâtes
- Amaverunt => ils aimèrent
=> mostly lost the "vi" part except first person.
- Finivi ====> je finis
- Finivisti ===> tu finis
- Finivit ===> il finit
- Finivimus => nous finîmes
- Finivistis => vous finîtes
- Finiverunt => ils finirent
=> also lost the "vi" part. Finire is Latin's forth conjugation.
- Potui ====> je pus
- Potuisti ===> tu pus
- Potuit ===> il put
- Potuimus => nous pûmes
- Potuistis => vous pûtes
- Potuerunt => ils purent
=> lost the "ot" and "i" parts.
- Attendi =====> j'attendis
- Attendisti ==> tu attendis
- Attendit =====> il attendit
- Attendimus ==> nous attendîmes
- Attendistis => vous attendîtes
- Attenderunt => ils attendirent
=> almost perfectly the same!