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A sense of oneness is a characteristic of the state of enlightenment. Whether this is Eastern-style enlightenment, or a Western-style peak experience, or another type of transcendental occurrence, we discover the single ground from which all phenomena emerge -- all people, all other living things, all physical objects, all energies. The phenomena's distinctions become secondary to the reality of their undifferentiated spiritual essence; i.e., the figure becomes the background.
In oneness, we might retain our individuality. The Buddha said that the individual no longer exists in this oneness; he or she is lost like a drop of rain falling into the sea. Other religions say, paradoxically, that the state of enlightenment grants us a knowledge of total oneness with spirit but also a retention of our individuality within that oneness. In this sense, enlightenment is a parallel to the "meeting with the Self," in which we transcend the ego (knowing that we are more than the ego) while still acknowledging the ego within its own realm and with its own still-valid functions; in fact, according to Jung, we encounter this Self only after we have developed the ego to its fullness.
Sometimes we try to establish other types of "oneness." Spiritual oneness is on a transcendental level, but human history is filled with the efforts to establish oneness in other realms -- political, military, religious, cultural, etc. Perhaps it is a perversion of our individual drive to experience transcendental oneness that is a reason for humanity's persistent desire to impose a type of oneness through such means as military world-conquest, and religious intolerance, and cultural imperialism, and political dictatorships, and other forms of monomania. (Of course, we can discern additional reasons for that drive, e.g., a world-conqueror's desire for wealth and power.) In the study of logic, this fallacy is called a "category error" -- trying to establish the qualities of one category (the realm of spirit) into another category (the realm of humanity). Some people dismiss these drives for material oneness; instead, they value diversity, freedom, and privacy in their human life.
We might feel oneness with groups. Instead of a oneness with all of creation, we might experience oneness with our own groups, e.g., our family, or our co-workers. This type of oneness is not based on a commonality of spiritual essence, but on a commonality of human identity, purpose, interest, or emotion. This type of oneness is likely to be a short-term condition while we are involved in a group activity (e.g., our job, or a sports event); it does not imply an overall wholeness in our relationship with the individual members. The oneness of group identity is useful in strengthening people's resolve and opportunity for cooperation, but it creates enemies. Group-oneness is intensified when the in-group feels threatened by the out-group; for example, citizens unite during wartime to fight a common enemy, thus increasing both our oneness (within our own group) and our "otherness" (toward the outsiders). As Sigmund Freud said in a letter to Albert Einstein, "The love of country has succeeded at bridging people at the national level. The great new historical challenge is the development of love among all the earth's inhabitants, and for the earth itself." In other words, we can love our group without hating other groups; instead, we can expand our sense of identity (and a type of oneness) to include ourselves, our family, our community, our nation, our humanity -- and the spirit-substance which underlies it all.