A Conjunx Endura is a spouse, another person who one has formed a strong emotional connection with. Though the Matoran species on Okoto have biological reproduction and relationships, the majority of other bio-mechanical words do not. A Conjunx Endura relationship is a relationship that is bonded by a ceremony called the Conjnx Ritus and praised by the community, much like the real life practice of Marriage.
This deep emotion bond is a very old practice that dates back to Hanahs time. The bond allows for the spouses to make decisions, mostly medical, for the other should they disabled or otherwise incapacitated.
Similarly is the Amica Impura, an officially bond like the Endura but is not based upon romantic connection. The equivalent of formal best friends, aka Best Friends Forever.
Both bonds are based on the Transformers phrases of the same meaning.