Short answers, pulled from the story.
In ancient Roman law and religion, sacramentum referred to an oath or vow that rendered the swearer sacer, meaning "given to the gods," in the negative sense if he violated it. It also referred to a thing pledged as a sacred bond, forfeit if the oath were broken. Both usages imply an underlying sacratio, an act of consecration.
The sacramentum established a direct relation between the swearer and the gods, while the iusiurandum was an oath of good faith within the human community, witnessed by the gods but operating inside human law. The iusiurandum was more common in legal applications such as swearing an oath in court.
The sacramentum militare was the oath taken by Roman soldiers pledging loyalty to the consul in the Republican era, or later to the emperor. According to Vegetius, soldiers swore to faithfully execute all that the emperor commands, to never desert the service, and to not seek to avoid death for the Roman republic.
Because the soldier had placed his life on deposit with the gods by swearing the sacramentum, he was considered sacer and subject to penalties including corporal punishment and execution that were inappropriate for civilian citizens, at least under the Republic. The same logic applied to gladiators, who swore their own version of the oath.
By the 3rd century, the sacramentum was administered annually on the 3rd of January, as attested by the Feriale Duranum, a calendar of state ritual discovered at Dura-Europos that dates to the reign of Severus Alexander (222-235 AD).
The English word "sacrament" derives from the Latin sacramentum. Apuleius used the word to refer to religious initiation, a usage that points directly to the transition in meaning from Roman oath to Christian rite.