Learn Finnish: Kiipeilijät tarkistivat turvaköyden. - The climbers checked their safety ropes.

This is an example of the distributive singular. Because each climber has exactly one safety rope of their own, Finnish uses the singular form. It essentially means the climbers checked their own individual safety rope.

What would it mean if we used the plural object turvaköydet instead?

If you wrote Kiipeilijät tarkistivat turvaköydet, it would mean the group of climbers collectively checked a general set or pile of ropes. Using the singular turvaköyden emphasizes that each person individually checked their own gear.

Why doesn't turvaköyden have a possessive suffix to translate the word "their"?

When ownership is obvious from the context, Finnish often drops the possessive suffix. You could add -nsä and say turvaköytensä, which is also correct, but simply saying turvaköyden sounds perfectly natural when it is understood that they are checking their own gear.

Why is the object turvaköyden in the genitive-accusative case ending in -n rather than the partitive?

The verb tarkistaa (to check) takes a total object when the action is completed. Because the climbers finished checking the ropes, we use the -n ending. If they were still in the middle of checking them, you would use the partitive form turvaköyttä.