"We do not love her."
Translation:Nie kochamy jej.
The verb "kochać" (to love) takes Accusative.
"ją" is the Accusative form of "ona" (she), and it translates to "her" as in "I love her".
"jej" is the possessive "her" as in "I like her dog".
"jej" is also the Genitive form of "ona".
If you negate a verb that normally takes Accusative, it takes Genitive instead.
In this example it's a different thing: "kochać" takes Accusative (Kochamy ją), but it's negated here, and if you negate a verb that takes Accusative, it takes Genitive instead (Nie kochamy jej).
But your question is still valid of course - br0d4 compiled a great guide on Genitive, including a whole list of verbs, here: https://forum.duolingo.com/comment/16569658
Fun fact (ok, not that fun maybe) - many native speakers use Accusative with some common verbs that should take Genitive.
The verb "kochać" takes a direct object in the Accusative case: "Kochamy ją" (We love her).
When a verb which took Accusative gets negated, it takes Genitive instead: "Nie kochamy jej" (We do not love her).
Don't take this rule too far, only Accusative changes when negated, other cases stay the same.
So yeah, both your sentences are ungrammatical.