Simply put, Serapis (Sarapis, Zaparrus) was an invented god. He was a composite of several Egyptian and Hellenistic deities who was introduced to the world at the beginning of the Ptolemaic (Greek) Period in Egypt during the reign of Ptolemy I, though his legacy lasted well into the Roman period. Thus, he was meant to form a bridge between the Greek and Egyptian religion in a new age in which their respective gods were bought face to face with each other, so that both Egyptians and Greeks could find union in a specific supreme entity. 

Linguistically, the god's name is a fusion of Osiris and the bull Apis, which by the Greek period might be said to have represented the essence of Egyptian religion. In fact, a cult of this combination god, named Osirapis (or Userhapi, Asar-Hapi), had existed in Egypt prior to the rule of the Ptolemies. Osirapis was basically the ssacred bull of Memphis after its death. According to the hieroglyphic texts which were found on stelae and other objects in the Serapeum at Sakkara, established long before the Greek Period, Apis is called "the life of Osiris, the lord of heaven, Tem {with} his horns {in} his head." and he is said to "give life, strength, health, to thy nostrils for ever."

Elsewhere from the 18th Dynasty, Osirapis is described as, "the great god, Khent, Amentet, the lord of life forever," Apis and Osiris were joined together by the priests of Memphis, where the attributes of Apis had been made to assume a funeral character and hence recognized as a god of the Underworld. On a monument of the 19th Dynasty, Apis is said to be "the renewed life of Ptah," and in an inscription of the 25th Dynasty he is called the "second Ptah." In the same text we have a mention of the "temple of Asar-Hapi (Osirapis)," and here it is clear that his identity had been merged with that of Osiris. The identification of Apis with Osiris was easy enough, because one of the most common names of Osiris was "Bull of the West". Apis was, in fact, believed to be animated by the soul of Osiris, and to be Osiris incarnate. The appearance of a new Apis was regarded as a new manifestation of Osiris upon earth.

However, the Greeks added to this Egyptian Core a number of Hellenistic deities, including Zeus, Helios, Dionysus, Hades and Asklepius to form Serapis. Eventually, these Hellenistic deities would predominate the god's final form. He then emerged as a supreme god of divine majesty and the sun (Zeus and Helios), fertility (Dionysos) the underworld and afterlife, as well as healing (Hades and Asklepius). However, his attributes regarding the afterlife and fertility were always primary to his nature.

Iconography

"In the city on the borders of Egypt which boasts Alexander of Macedon as its founder, Sarapis and Isis are worshiped with a reverence that is almost fanatical. Evidence that the sun, under the name of Sarapis, is the object of all this reverence is either the basket set on the head of the god or the figure of a three-headed creature placed by his statue. The middle head of this figure, which is also the largest, represents a lion's; on the right a dog raises its head with a gentle and fawning air; and on the left the neck ends in the head of a ravening wolf. All three beasts are joined together by the coils of a serpent whose head returns to the god's right hand which keeps the monster in check."

Macrobius, Saturnalia (I.20.13)

The iconography of Serapis was dominated by Hellenistic elements. In his anthropomorphic form, he was represented as a man wearing a Greek style robe with a Greek hairstyle and full beard. Surmounting his head was often a basket or a tall, dry corn measure (holding a quarter of a bushel), representing his fertility attributes as well as his association with Osiris, who was sometimes a god of grain. At times, he was also provided with curved ram's horns. At his feet might also set the three-headed dog Cerberus, the guardian of the underworld.

Occasionally, and particularly in conjunction with his consort Isis, the greatest Egyptian goddess during the Greek Period, both deities could be depicted as serpents with human heads (most often on door jambs), where Serapis would be discernable by his beard. When represented in such a fashion, it was usually in relationship to their aspects related to the netherworld and fertility. 

The chief center of the worship of Serapis in Ptolemaic times was Alexandria at the great Serapeum, which was considered a wonder and a site of pilgrimage throughout the Mediterranean world, until it was destroyed by order of Emperor Theodosius in 389 AD.  The Serapeum which Ptolemy repaired, or founded, was probably around Rhakotis near Pompey's pillar and was a very remarkable building. 

Interestingly, Rhakotis was the small Egyptian village that had been located on the site of what would become Alexandria, and some traditions hold that Osirapis was its local God.

The Temples main plan seems to have resembled that of the famous Serapeum at Memphis, but parts of it were richly painted and gilded, and it possessed a fine library which was said to contain some 300,000 (or perhaps as many as 42,000) volumes. The library was actually an annex of the Great Library of Alexandria, and hence known as the "Daughter Library".

Within the temple was a specific and famous statue of Serapis. How the statue came to be in Alexandria at this temple is of some interest. Tradition holds that, while Ptolemy was considering the possibility of a hybrid god to unit the Egyptians and Greeks, he had a dream, wherein a colosssal statue of some god appeared to him and bid the king to remove it to Alexandria. According to Plutarch (De Iside et Osiride, 28), he had never seen a similar statue, and he knew neither the place where it stood, nor to whom it belonged. One day he happened to mention his dream to Sosibius, and described the statue which he had seen, whereon this man declared that he had seen a statue like it at Sinope. Tradition says that this was Sinope on the Pontus, and adds that as the inhabitants of the city were extremely unwilling to part with their statue, it, of its own accord, after waiting for three years, entered into a ship and arrived at Alexandria safely after a voyage of only three days. However, others provide that after three years of futile negotiations, Ptolemy's men simply stole, the statue, claiming that it had boarded their boat on its own. 

There were other smaller temples and shrines dedicated to this god in various locations throughout Egypt, but the god's cult was also spread throughout much of the Graeco-Roman world by traders and other converts. Another Notable cult center was the Greek holy site of Delos, which was founded by an Egyptian priest in the third century BC. 

There was even a Roman Period sculpted head of Serapis, dating to the second or early third century AD, discovered in London at the Walbrook Mithraem, and a temple of Serapis is mentioned in an inscription found at the Roman site of Eburacum (modern York) in the United Kingdom. Hence, he was even important enough to reach the most distant areas of the Roman Empire.

Interestingly however, Serapis really never received wide acclaim in Egypt itself, where other more traditional Egyptian deities continued to receive more popular worship.

Early Christianity

Serapis may have finally had certain ties with the early Christian community. There were certainly some similarities between Serapis and the Hebrew God. Serapis was a supreme god,  and it seems that some early worshippers of Christ amongst the Gentiles could have possibly worshipped Serapis either purposefully, or confusing him with Christ, though the confusion seems more likely to have been one of language.

A correspondence of Emperor Hadrian refers to Alexandrian worshippers of Serapis calling themselves ‘Bishops of Christ’:

'Egypt, which you commended to me, my dearest Servianus, I have found to be wholly fickle and inconsistent, and continually wafted about by every breath of fame. The worshipers of Serapis (here) are called Christians, and those who are devoted to the god Serapis (I find), call themselves Bishops of Christ.'

–Hadrian to Servianus, 134A.D. (Quoted by Giles, ii p86)

In fact, it appears that some followers of Serapis were eventually expelled from Rome when, in 19 AD, Tiberius also expelled the Jews. 

Nevertheless, how great confusion between Serapis and Christ could have existed is really somewhat questionable. In 68 AD, a mob of pagans is said to have formed at the Serapis Temple in Alexandria, who then descended on the Christians who were celebrating Easter at Baucalis. There, they sized St. Mark, dragging him through the streets, before throwing him in prison. Clearly those worshippers of Serapis and Christ were aware of each other and the differences within their religions, though perhaps at a later date, some amongst the worshippers of either may have chosen to cover all of their options.

On the other hand, some have pointed out that Chrestus (Christus) was another name for the Egyptian god, Serapis. Chrestus may be translated as "Messiah", though the term need not apply to any specific Messiah, such as Jesus. It therefore could have simply been applied to "Lord Serapis", so that in fact, there was never any connection at all between the early Christians and the worshippers of Serapis. 

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