Learn Russian: В партере нет ни одного свободного места. - There isn't a single empty seat in the stalls.

The word нет (there is no) always requires the genitive case for the thing that is absent. Because the noun место (seat/place) is neuter, the emphatic pronoun ни один takes the neuter genitive form ни одного, and the adjective becomes свободного.

Why do we need the particle ни here? Couldn't we just say нет одного свободного места?

Russian requires double negation for this structure. Adding ни to одного creates the emphatic meaning 'not a single.' If you dropped the ни, the sentence нет одного свободного места would literally mean 'one empty seat is missing' (meaning you are exactly one seat short), rather than 'there are zero empty seats.'

What exactly is a партер?

It is a theater term borrowed from French. In a Russian theater, the партер refers to the stalls or the seating area on the main ground floor of the auditorium, directly in front of the stage.

Can I change the word order and put нет at the end of the sentence?

While Russian word order is flexible, putting нет at the very end (В партере ни одного свободного места нет) places heavy, dramatic emphasis on the word 'none' or 'is not.' The current order is the most natural, standard way to state this fact.