Deus ex machina is a Latin expression that literally translates to "god from the machine". It is a plot device where an unsolvable problem is immediately resolved by an unforseen and unlikely event. It is often criticized as a cheap way of resolving a plot to reach its desired end. An example of this technique is an ending of a plot where the events may just have been a product of a character's dream.
The expression deus ex machina derives from Ancient Greek and first appeared in Greek tragedies, where a divinity would appear thereby resolving the situation. In Greek tragedies, an actor portraying a god would enter on a machine, usually a trapdoor, and resolve the plot. From thereon, deus ex machina developed to include events or characters that simply resolved the plot in an unforseen and abrupt manner.
The correct pronunciation of deus ex machina in Latin is deh-oos ex mahk-ee-nah. You will note that the vowels are much softer in Latin than they are when deus ex maxchina is pronounced by English speakers. To wit, deus is often pronounced as day-us or machina is pronounced as mah-kin-uh. However, this is imprecise. You should look to pronounce the vowels in a softer way to achieve an impressive pronunciation of deus ex machina. Therefore, the "e"s are pronounced with an "eh" sound, the "a"s with an "ah" sound, the "u" with an "oo" sound and the "i" as an "ee" sound.