Theism is broadly defined as the belief in the existence of either a deity or deities. In popular parlance, or when contrasted with deism, the term often describes the classical conception of god(s) that is found in the monotheistic and polytheistic religions; a belief in a god or in gods without the rejection of revelation as is characteristic of deism. 
The term theism derives from the Greek theos meaning "god". The term theism was first used by Ralph Cudworth (1617-1688). In Cudworth's definition, they are "strictly and properly called Theists, who affirm, that a perfectly conscious understanding being, or mind, existing of itself from eternity, was the cause of all other things".
Atheism is commonly understood as rejection of theism in the broadest sense of theism, i.e. the rejection of belief in a god or gods. The claim that the existence of any deity is unknown or unknowable is agnosticism.
Monotheism (from Greek ?????) is the belief in theology that only one deity exists. Some modern day monotheistic religions include Christianity, Judaism, Islam, Baha'i Faith, Sikhism, Zoroastrianism, Eckankar and some forms of Hinduism.
Polytheism is the belief that there is more than one god. In practice, polytheism is not just the belief that there are multiple gods; it usually includes belief in the existence of a specific pantheon of distinct deities.
Within polytheism there are hard and soft varieties:
Polytheism is also divided according to how the individual deities are regarded:
Some people find the distinction between these two beliefs as ambiguous and unhelpful, while others see it as a significant point of division. Pantheism may be understood a type of Nontheism, where the physical universe takes on some of the roles of a theistic God, and other roles of God viewed as unnecessary.
Deism typically rejects supernatural events (such as prophecies, miracles, and divine revelations) prominent in organized religion. Instead, Deism holds that religious beliefs must be founded on human reason and observed features of the natural world, and that these sources reveal the existence of a supreme being as creator.
Autotheism is the viewpoint that, whether divinity is also external or not, it is inherently within 'oneself' and that one has the ability to achieve godhood. This can be in a selfless way, a way following the implications of statements attributed to ethical, philosophical, and religious leaders (such as Jesus and Mahavira).
Autotheism can also refer to the belief that one's self is a deity, within the context of subjectivism. Hindus use the term, "aham Brahm?smi" which means, "I am Brahman". This is a fairly extreme version of subjectivism, however.
Atheism, in general, the critique and denial of metaphysical beliefs in God or spiritual beings.... Instead of saying that an atheist is someone who believes that it is false or probably false that there is a God, a more adequate characterization of atheism consists in the more complex claim that to be an atheist is to be someone who rejects belief in God for the following reasons (which reason is stressed depends on how God is being conceived)...
On our definition, an 'atheist' is a person who rejects belief in God, regardless of whether or not his reason for the rejection is the claim that 'God exists' expresses a false proposition. People frequently adopt an attitude of rejection toward a position for reasons other than that it is a false proposition. It is common among contemporary philosophers, and indeed it was not uncommon in earlier centuries, to reject positions on the ground that they are meaningless. Sometimes, too, a theory is rejected on such grounds as that it is sterile or redundant or capricious, and there are many other considerations which in certain contexts are generally agreed to constitute good grounds for rejecting an assertion.(page 175 in 1967 edition)
In the most general use of the term, agnosticism is the view that we do not know whether there is a God or not.(page 56 in 1967 edition)
In the popular sense, an agnostic is someone who neither believes nor disbelieves in God, whereas an atheist disbelieves in God. In the strict sense, however, agnosticism is the view that human reason is incapable of providing sufficient rational grounds to justify either the belief that God exists or the belief that God does not exist. In so far as one holds that our beliefs are rational only if they are sufficiently supported by human reason, the person who accepts the philosophical position of agnosticism will hold that neither the belief that God exists nor the belief that God does not exist is rational.