This is one version of an argument for the existence of God, often called the Cosmological Argument. One might object to the argument in various ways. But at its heart, there is an enormous logical fallacy. The sentence ‘Everything has a cause’ is ambiguous. It can mean that everything that happens has some cause or other - that is, for every x, there is a y, such that x is caused by y; or it can mean that there is something which is the cause of everything - that is, there is some y such that for every x, x is caused by y. (36-7)Introductory logic books generally have a section on how to identify an argument. Whatever argument this is, though, it's not the cosmological argument. Priest is committing an informal fallacy here--the straw man--which his Logic, interestingly, omits.
The only people who identify the "everything has a cause" argument are thinkers ignorant in the history of philosophy. (Bertland Russell, Sam Harris, Daniel Dennett and Richard Dawkins attack this straw man in Why I Am Not A Christian, Letters to a Christian Nation, Breaking the Spell, and The God Delusion respectively.) In fact the "everything has a cause" argument seems to have originated from such second-rate thinkers (with the exception of Russell) and have been transmitted through their writings.
Readers who want to see what the various cosmological arguments actually say can consult the Stanford Encyclopedia entry.