Monotheism: one God, and no others. Judaism, Christianity, Islam, Sikhism, Zoroastrianism in its own self-description, and Baha'i. What counts as a violation is internally disputed, and both the Christian Trinity and the veneration of saints have been charged by other monotheists with compromising it, which shows that the category's boundaries are argued over from inside rather than being self-evident.
Polytheism: many gods, distinct and individually worshipped. Historical Greek, Roman, Norse, Egyptian and Mesopotamian religion, and much of Hinduism and Shinto. Worshippers of many gods generally do not regard them as interchangeable or as aspects of one another, and the assumption that polytheism is a confused or immature monotheism is a legacy of polemic rather than a description of practice.
Henotheism: the worship of one god at a time as supreme, without denying that others exist. Max Muller coined the term for the Vedic pattern in which whichever deity is being addressed receives the language of supremacy, and it names something real in a good deal of Hindu devotional practice, where a devotee's chosen deity is the highest for them.
Monolatry: the worship of one god while accepting that others exist. Historical-critical scholarship widely holds that early Israelite religion was monolatrous before it became monotheistic, pointing to texts that presuppose other gods' existence while forbidding their worship. More traditional interpreters contest the reading, and the disagreement is a good example of the divergence between confessional and historical-critical approaches.
Pantheism: God and the totality of what exists are identical. Spinoza is the classic case. Some Advaita-influenced and Stoic thought is placed here, though Advaitins generally reject the label, since the world's status in Advaita is not straightforwardly real and identifying Brahman with it misdescribes the position.
Panentheism: the world is in God, and God exceeds it. Various Hindu, Kabbalistic, process and modern theological positions, offered as a middle way preserving both transcendence and genuine relation.
Animism: the attribution of spirit or personhood to a wide range of beings and features of the world. E. B. Tylor coined it in 1871 as a theory of religion's origin, a use now abandoned. The "new animism" of Irving Hallowell, Nurit Bird-David and Graham Harvey has recovered the word to describe relational ontologies in which persons are not only human, and it is used in that sense by many Indigenous scholars.
Animatism: R. R. Marett's term for belief in an impersonal power rather than in personal spirits, with Melanesian and Polynesian mana as the standard example. The concept of mana has itself been substantially reinterpreted since, and the early accounts are now regarded as distorted by the theoretical expectations of those collecting them.
Pandeism: a minority philosophical position combining pantheism and deism, holding that God became the universe and does not persist as a separate consciousness. It has essentially no institutional presence and is chiefly a philosophical proposal.