De ober zet het warme bord op tafel en controleert onze bestelling.

In Dutch every noun is either “common gender” (using de) or “neuter gender” (using het). There are no strict rules that always tell you which is which—you need to learn them along with each noun.
• de ober, de tafel, de bestelling = common gender → de
• het bord = neuter gender → het

Because when an adjective comes before a noun with a definite article (de/het), it almost always takes an -e ending. That’s called adjective inflection:
• het + warm + bord → het warme bord
If there were no article—een warm bord—the adjective stays in its base form after een for neuter nouns.

Here it’s simply the verb zetten (“to place/set [upright]”) followed by the preposition op indicating location.
• zet + op tafel = “places on the table”
opzetten (separable verb) is a different verb meaning “to set up,” “to switch on,” etc.

The sentence uses coordination with en (“and”) and omits the repeated subject in the second clause because it’s the same as in the first clause. This is called subject ellipsis:
“De ober zet … en (hij) controleert …” → “De ober zet … en controleert …”

Dutch possessive pronouns agree with the gender of the noun they modify:
• bestelling is a de-word (common gender) → use onze
• If it were a neuter noun (het-word), you’d use ons