Theism is the traditional belief that God is a being, with a personality and mind, that stands outside of our own universe and yet exerts some amount of control and power over the physical world (perhaps as creator, and continued sustainer). Usually theists also believe that God is all-good, all-powerful and all-knowing. They key phrase here is that God is both a being and personal (having a personality). There are beings that are not personal, at least as far as we can tell, such as bacteria, algae, and so on. More on that when I get to deism.
Technically, atheism is the denial of theism, with the prefix a- meaning not, as in not a theist. What an atheist denies is that God has this particular package of divine traits. But it does not mean that the person does not believe in God at all. Although some people have tried to create a newer word -- nontheism, for religious beliefs that don't accept a personal God -- there was no need to. At its base, this is what atheism means. Unfortunately the term atheism has been kidnapped from its original meaning to mean the rejection of any religious position, and even worse atheism became associated with people who have no morals along with no religion. The term nontheism came to use because non-religious folk were afraid of being misunderstood when they said that they were atheistic. A person can be quite moral, and actually religious while being an atheist in the truest sense of sense of the word.
However, in defense of the way the word usage has evolved, in my experience most atheists are actually rejecting the dominant religious view of their culture (such as rejecting traditional Christianity) and are wanting to say that they are anti-religious. They often site the fact that God cannot be all-loving and all-powerful and allow evil to exist. Many never question whether God might have a different set of characteristics. Technically, one can be a theist and believe that God is evil, angry and wants to make your life miserable; rendering that argument insufficient for the denial of theism. Such people are more accurately antitheists, they are rejecting and standing strongly against the religious tradition, and oppose the belief in God.
Other people who use that label atheist will describe themselves as believing in science, in what is observable, repeatable, and arrived at by means of logic and research. Technically this is not atheism either, because there are many theistic possibilities that do not conflict with science. What these people are really saying is that they reject the fundamentalist position on God creating the world in seven days, or some other specific doctrine held by a religious sect (such as the earth being flat, or the sun orbiting the earth, or the idea that heaven resides on the dark side of the moon.) Such persons have a black and white view of religions, you either believe it all literally or none of it; and probably could be helped by reading further about other ways of viewing God.
Deism is the belief that God set the universe in motion, but then left it to its own. People often refer to this as the divine watchmaker theory. God is that which initially started the big-bang, and then stepped back to watch the show. Philosophers who suggest that our lives may actually be part of a computer simulation done by a higher being, are in a sense deists. The programmer is God, and we are simply the bits of code doing what the program tells us. Deism is usually theistic as in the example above, but not necessarily so. A deist could believe in a non-personal God setting things in motion by a natural life process, and simply not knowing nor caring that we exist as a product of its own life.
Panentheism (which my spell-checker doesn't even recognize as a word) is the belief that God and the universe are actually linked. Many people say that a Panentheist is a person who believes that the universe is the body of God, the physical manifestation of God, while God's mind or spirit are either outside of the universe or the consciousness of the universe. The idea parallels the philosophy of a mind-body dialectic, we are physical and yet have consciousness, why couldn't the universe also have a larger consciousness. Panentheism often gets confused with Pantheism, as evidenced by one of my favorite scenes from Britcom Red Dwarf.
Kryten: Surely you believe that God is in all things? Aren't you a pantheist?
Lister: Yeah, but I just don't think it applies to kitchen utensils. I'm not a FRYING pantheist. Machines do not have souls. Computers and calculators do not have an afterlife. You don't get hairdryers with tiny little wings, sitting on clouds and playing harps.
What is being described here is more like Panentheism -- God in all things, and yet beyond.
Transtheism is similar to Panentheism because it believes that gods were once physically part of the universe, but have ascended into a spirit world which is outside of what we know. Most religions that include a worship of ancestors are forms of transtheism. In fact, within Christianity the veneration of the saints is close to transtheism, although its practitioners would argue that they are only asking the saints to intervene with God on their behalf -- there still is the reverential approach to that which was once physical and is now spirit.
Pantheism is the belief that nature is all that there is. In other words, God is nature, nature is God. God does not have personality, other than the personalities that exist in nature. God is no more spirit than nature is spirit. God is the universe and the universe is God. Many people who call themselves atheists are in my opinion more accurately described as pantheists. They wouldn't call themselves that because they don't think they worship nature, but if they believe that nature is the only reality and only real power -- in my opinion that is what they actually are.
Others who claim the tag of atheist could also be called humanist. They still believe that there are ethics and morals that guide our lives, and that our lives are to be lived out within that code. It also usually indicates a value for humanity, individually and as a community; and prefers freedom of thought over doctrine and faith. Unfortunately the term humanist is as vague as the term atheist, and it has been used for so many groups that are religious in nature, as well as those that are not. In general, humanism is not a theological statement; but an ethical one.
Agnosticism is when a person isn't sure what they believe. Usually though it refers to a person who isn't sure if there is a theistic God. There may be or there may not be. Many intellectual people who have struggled with traditional church doctrine are agnostics. Honestly, most Christians and pastors at some time or another have moments of agnosticism. An agnostic may be considering all of the above as possibilities and simply admitting that it is difficult to tell the truth about the mystery of God.
Here's the deal -- deists, panentheists, frying pantheists, some humanists, and pantheists are all atheists because they reject the theistic view of God. So, all you atheists really should pick a more specific word that describes your beliefs -- it would be clearer, and you may discover you have more in common with many religious folk than you think, because I know practicing Christians who are all of the above.