Learn Russian: Мы не понимаем обсуждаемый термин.

It comes from the imperfective verb обсуждать (to discuss). To form a present passive participle, you take the present tense stem and add -ем- (for most type 1 verbs), followed by an adjective ending. Here, it matches the masculine noun термин, giving us обсуждаемый.

Since обсуждаемый термин is the direct object of the verb понимаем, why doesn't its ending change?

It actually is in the accusative case! Because термин is a masculine inanimate noun, its accusative form is identical to its nominative form. The participle agrees with the noun, so it keeps the standard -ый ending instead of taking the animate -ого ending.

The English translation says 'the discussed term'. Does the present participle обсуждаемый mean the discussion is happening right now?

Yes, Russian present passive participles indicate an ongoing action. A more literal translation is 'the term being discussed'. If the discussion were already finished, Russian would use a past passive participle instead.

How would a Russian speaker normally say this in casual, everyday conversation?

While participles are great for written or formal Russian, in everyday speech people usually replace them with a который (which) clause. You would likely hear: Мы не понимаем термин, который мы обсуждаем.