Learn Russian: Президент хочет, чтобы они закончили официальные переговоры. - The president wants them to end the official negotiations.

This is the core rule of the чтобы structure when there are two different subjects. The president (Subject 1) wants them (Subject 2) to do something. In Russian, after чтобы, you must always use the past tense for the second verb, regardless of when the action actually takes place.

English uses an infinitive here (wants them to end). Can I just say хочет их закончить?

No, that would mean 'The president wants to finish them' (meaning the president is doing the finishing, and 'them' is the object being finished). When the person wanting the action is different from the person doing the action, Russian requires a two-part sentence: хочет, чтобы + Subject 2 + past tense verb.

Why use the perfective verb закончили instead of the imperfective заканчивали?

The president wants them to actually complete the negotiations and achieve a final result. Perfective verbs like закончили focus on the successful completion of an action. If we used the imperfective заканчивали, it would mean he wants them to spend time in the ongoing process of wrapping things up.

What case is официальные переговоры in?

It is in the accusative case because it is the direct object of the verb закончили (what are they ending?). Since переговоры (negotiations) is an inanimate noun, its accusative plural form looks exactly identical to its nominative plural dictionary form.