Learn Hardcore Russian: Моя компания купила новое оборудование.

Аппаратура: often refers to electronic or technical apparatus (e.g. lab or broadcast equipment).
Техника: covers machinery, appliances, technical devices (often consumer or industrial machinery). Оборудование is the broadest term, covering any set of machines, tools, installations used for work or production.

In Russian past-tense verbs agree with the subject’s gender and number. Компания is a feminine noun, so the past tense of купить takes the feminine ending -ла, giving купила. If the subject were masculine (e.g. директор), it would be купил.

Why does новое end in -ое, and how does it agree with оборудование?

Новое is the neuter singular form of the adjective новый in the nominative/accusative case. It must match оборудование, which is a neuter singular noun ending in -ие. In Russian adjectives always agree with their nouns in gender (masculine/feminine/neuter), number (singular/plural), and case.

Which case is оборудование in, and why doesn’t its ending change?

Here оборудование is the direct object of the verb, so it’s in the accusative case. Neuter inanimate nouns have identical forms in nominative and accusative, so you don’t see a change in the ending.

Why aren’t there words for “the” or “a” before компания or оборудование in Russian?

Russian has no articles (no direct equivalents of “a,” “an,” or “the”). Definiteness or indefiniteness is understood from context. So “a company” or “the company” is just компания.