Syncretism is a process that generally happens when two polytheistic faiths come in contact with one another. It is the revelation of two or more deities being one in the same based on mythology, cult worship, and dominion. This was a very common occurrence, especially in the areas under Greek (Greco-Egyptian) or Roman (Gallo-Roman) influence.
Herodotus identified several Egyptian deities with Greek ones in his Histories. Most famously you see Thoth being equated to Hermes, Amun being equated to Zeus, and Osiris being equated to Dionysos. The Greco-Egyptian pantheon was a melding of Greek and Egyptian ideas based on the similarities found in both religions. It was generally seen in the port cities like Alexandria, and it picked up a following during the Ptolemaic dynasty.
Also in Greece you can see an adoption of deities from the Thrace, Asia Minor, and several other places. Deities thought to have been adopted from other peoples include Hekate, Dionysos. There are two traditions that suggest Dionysos is not of Greek origin. The first is from Herodotus, who says that Melampus brought the cult to Greece from Egypt. The second traditions states that Orpheus, a Thracian, brought the cult to the Greeks. Other bits to consider include, Semele being a non Greek name, Dionysos association with leopards, and his travels to India. Leopards and India might not seem like that big of a deal, but if you look at Hinduism, specifically the god Shiva, you'll see the significance. Shiva is more or less the spitting image of Dionysos except for skin color. (Shiva has blue skin) Even the cult practices and associations are the same for the two deities. Honestly though, this shouldn't come as a surprise since the Indians and Europeans share common ancestry and thus common religious practices. Trade between the two cultures would have helped preserve the similarities.
Roman syncretism happened a lot as Rome came in contact with the Greeks and northern tribes (Germanic and Gallic). Anyone who has studied mythology or history will tell you about the striking similarities between the Roman and Greek faiths. But contrary to popular belief the two are not one in the same. The Religio Romana is very different from Hellenism. The mythologies are the same but the worship and view of the Gods is not. The Romans had a religion before they came in extensive contact with the Greeks. Romulus and King Numa are the supposed founders of the Roman faith. The religion centered around numina (spirits) that ruled every aspect of life. These spirits were not depicted in art since all human creations are impermanent and the numina were not. Once the Romans began to trade and conquer Greek cities there was a religious shift. The old laws gave way to new ones and the numina were given faces, the faces of the Olympians. The Romans identified many of their deities with those of Greece and thus the misconception was born. Greco-Roman syncretism was so perfect and flawless that many people believe what I stated earlier, that they are one in the same.
Julius Caesar and Tacitus are famous for writing things like "The whole of Gaul worships Mercury." These claims are just proof of the the global view of deities in ancient Rome. Obviously the Gauls didn't worship Mercury per say, but they did worship a god so similar that they were thought of as one. Tacitus also wrote of a mother goddess in Germania that he identified with Isis. He wrote extensively about the cult practices surrounding that mother goddess and how they were more or less the same as the Egyptian cults of Isis. Also in Gaul, as well as other part of the empire, you see adoption.
Mithras and Sol Invictus were adopted form Persia, and Isis was adopted from Egypt. These deities were given their own priesthood and temples in Rome and Roman controlled lands. Lastly you see foreign deities being incorporated into Roman mythology. In Gaul Epona, a tribal horse goddess, became a companion and possibly wife of Mars. Things like this were very common throughout the ancient world.
Modern syncretism is frowned upon by some circles. Usually the people who view it negatively are fundamental reconstructionist but that's beside the point. We shouldn't shun away form this practice. However, at the same time we need to be cautious of it. Eclecticism is the result of pagans going all willy nilly with syncretism. An example of bad syncretism would be a Heathen doing a Roman ritual for Kwan Yin. Obviously those three faiths shouldn't be mixed like that. Extensive research should be done before you equate one god to another. Even more research should be done before you design a ritual to honor said god. And you should always honor the deity in accordance to their name. Instead of doing a Roman style ritual for an Asian deity, do an Asian one. Even the ancients new this. Roman rituals to a Gallic god were in the Gallic style. What made them Roman were the people who participated, the language used, and a few ritual tools used.
When I employ syncretism I state the foreign deity (Freyr) and then the Hellenic/Roman deity (Liber/Dionysos). I also am sure to make the ritual suit the practices surrounding the foreign deity. In this case I utilize a drinking horn, bells, Futhark runes, etc This is a sign of respect to the culture, heritage, and deity. You may be wondering why does it matter which culture I match with which name if they are the same deity? Think of it like this, if someone calls out your name and then tries to have a conversation with you in a foreign language would you understand it? Probably not, and the same thing applies to rituals and the Gods. The phrase "When in Rome do as the Romans do" pops into mind.