The French loanword bourgeois works as both an adjective and a noun. Its main definition is of, relating to, or typical of the middle class, but it also works as a noun referring to a middle-class person. It often has negative connotations; as an adjective, bourgeois can be synonymous with conventional and conformist, which are often meant negatively when applied to people.
Bourgeoisie, which is only a noun, is more general. It denotes the middle class as a whole, rather than just a middle-class person.
The words are usually pronounced in the French manner—boor-ZHWA and boor-zhwa-ZEE.
Bourgeois is usually an adjective, and it’s usually negative or at least slightly scornful—for example:
The liberals of San Francisco want to build a bourgeois republic wherein practically perfect people live aesthetic, ethical lives. [Telegraph]
Yes, I am fully aware of the fraught nature of complaining about the loss of a $400 stroller, one that epitomized privilege, and all that is loathsome about urban bourgeois parenting to begin with. [New York Times]
Canadians are so bourgeois, so unencumbered by any real problems, that they actually riot over entertainment. [CBC]
And bourgeoisie is always a noun—for example:
Significant sections of the Syrian population, including the bourgeoisie from the key cities of Damascus and Aleppo, have yet to shift sides. [Sydney Morning Herald]
If the bourgeoisie does start to protest, the party will be faced with an old dilemma: liberalise or step up repression. [The Economist]