Short answers, pulled from the story.
Mystery religions were religious schools of the Greco-Roman world in which participation was reserved for initiates called mystai. Their defining characteristic was strict secrecy about the details of initiation and ritual practice, which could not be revealed to outsiders. They formed one of three types of Hellenistic religion alongside the imperial cult and philosophical religions such as Neoplatonism.
On the 15th of the month of Boedromion, as many as 3,000 prospective initiates gathered in the agora of Athens, then traveled a 15-mile procession to the sanctuary of Demeter and Persephone in Eleusis. The actual initiation took place at sunset in the Telesterion, a great hall, and may have spanned two nights, culminating in the display of a statue of Demeter, an ear of wheat, and a symbolic birth of agricultural wealth.
Mithras was a deity adapted for Roman purposes from the Zoroastrian god Mithra, retaining his Phrygian cap and garments as a marker of eastern origins. Worship of Mithras was particularly concentrated among Roman soldiers, originating in the 1st century BCE. The cult's underground initiation rites, hierarchical grade system of seven ranks, and emphasis on a handshake-based bond of membership appealed to military culture.
The seven grades of Mithraism were, in ascending order: Corax (raven), Nymphus or Gryphus (bridegroom), Miles (soldier), Leo (lion), Perses (Persian), Heliodromus (sun-runner), and Pater (father). Members who completed initiation were called syndexioi, meaning those united by the handshake.
Scholars have debated this question since at least the 19th century. Arthur Nock's 1952 article noted the near absence of mystery terminology in the New Testament, and research showed that Christian baptism derives from Jewish purificatory ritual rather than mystery cult practice. Beginning in the 4th century, Christians did adopt the Greek word mysterion to describe their own sacraments, using the concept of secrecy around rites without borrowing the ritual content.
The Samothracian Mysteries were the second most famous mysteries of the ancient world. Initiation ran from April to November and centered on anonymous deities known as the Samothracian gods or the Great Gods. Philip II of Macedon and his wife Olympias were said to have met during the initiation ceremony there. Initiates received a purple fillet and a Samothracian ring, a magnetic iron ring coated in gold.