Learn Russian: В нашем парке нет зверей.

The base word is зверь (animal or beast). Because it ends in a soft sign (-ь), it takes the -ей ending in the genitive plural. We need the genitive case here because the sentence uses нет (there is/are no).

I learned животное for animal. What is the difference between животное and зверь?

Животное is the broad, everyday word for any animal, including pets. Зверь is more specific and usually translates to a wild animal or beast. You will often see звери used for wild mammals in nature or at a zoo.

Why is наш spelled as нашем in this sentence?

It is in the prepositional case to match парке. Since the sentence describes a location using в (in), both the masculine noun парк and the pronoun describing it (наш) must change to their prepositional forms.

Why is there no Russian word for "there" at the start of the sentence?

Russian doesn't use placeholder words like the English "there" in "there is" or "there are". To say "there are no animals in our park", you simply state the location (в нашем парке) followed by нет and the genitive noun (зверей).