What does "Ex Machina" mean?

Edited due to many comments providing advice:

Deus Ex Machina doesn't originate from the (rather spectacular) video game Deus Ex.

Modern meaning

Nowadays, it's a simple phrase that exists in the English language and is defined as such:

a character or thing that suddenly enters the story in a novel, play, movie, etc., and solves a problem that had previously seemed impossible to solve

However, a slightly older meaning is:

a god introduced by means of a crane in ancient Greek and Roman drama to decide the final outcome

Latin translation

The slightly older meaning mentioned represents the literal Latin translation of Deus Ex Machina, meaning god from the machine. There is a Wikipedia article describing this in much more detail if you are interested, but in short, an actor playing a god or goddess would be lowered on stage by a "mechane" which was the name of the crane device used.

Typical usage

In modern films, a deus ex machina typically refers to some sort of character or plot device that solves all the issues at the end of a film (e.g. that everything was a hallucination).

Other interpretations follow the typical Greek usage and use the phrase to refer to the introduction or presence of a god or goddess.

Linkage to Deus Ex video game

There is no relationship between the film and the game you describe. Deus Ex is called so because of its relationship to the deus ex machina usage already alluded to.

Ex Machina film usage

If we remove "Deus" from "Ex Machina", this leaves "from the machine" [thanks to @ghostdog for pointing this out].

This quite literally describes Ava. She is from the machine—a creation. This perfectly fits the literal translation of the phrase.

It is also arguable that her creation by Nathan fits the literal "deus" part of "deus ex machina". He is the god, who has created from the machine something that can imitate reality. Whilst @Richard's answers show interviews with the directors of the film suggesting the former is all they were targetting, it's still an interesting interpretation.

_{Note: Reference to Bustle article has been removed as I felt I was unclear in how it linked into the rest of the answer}

Alex Garland spoke to this in an interview with TheFilmStage. The title quite literally means "The Machine" since ... er... that's what it's about, a machine.

TFS: I’ve read quite a few interviews with you about this film but I’ve yet to really see anyone ask you about the title itself. Ex Machina. Taken from Deus Ex Machina, so all it means is The Machine?

AG Yes, precisely. If you take the prefix, the Deus Ex, it implies the god out of the machine.

TFS: I think a title is incredibly important.

AG: I agree, yeah.

TFS: Did you get any pushback on the title?

AG: We did indeed. There were some people that felt the title was a bad idea. It’s not well known. People don’t know how to pronounce it. I really liked this title but I don’t have the power to fight for it that hard. A couple of people involved in the film, with Scott Rudin in particular and one of the financiers, just decided that it was the right thing to do and to support the title. The evidence that was used that finally managed to convince people to go with the title was the movie Prometheus. They figured if they could make Prometheus work, people would buy into Ex Machina. But in truth, if I’m being totally candid, the fact is that this is a really low-budget film and probably they just didn’t care that much.

He also spoke a little about the wider implications in this interview with IO9

We asked Garland if this movie was intentionally about an abusive childhood, and the idea that we’ll raise A.I.s the same way we raise human children, abusively. “That’s a really complicated question, and it’s got a complicated answer,” said Garland.

“Basically—if I can go back a step—you might know that Ex Machina comes from a larger phrase, which was “deus ex machina,” and the deus bit of that is God. And this title drops ‘God’ out of it. And some of my thinking ran along the lines of this... We typically present creation stories as cautionary tales, saying ‘Man should not meddle in God’s work. And I wasn’t interested in the ‘God’ part of it. So hence taking ‘God’ out of the title.”

Deus ex machina is a very old theatrical plot device, where the characters of gods would be brought on stage to develop the plot. It came to mean any 'outside influence' on the plot structure, which could not have been performed by the characters themselves.

Its literal meaning "God from the machine" has been reinterpreted over recent years, as the plot device is now almost too well-known to really use any more.

I think the movie's use of it assumes the audience to already be familiar with the concept, thereby allowing them to push one step further & just use "from the machine" to describe Ava's 'potential' as a human, rather than machine - as the entire plot device for Ava's sentience is Nathan's Turing test on successive Avas.
Caleb's character is the device by which we discover this.

The relationship between Ex Machina and Deus Ex is that both are based on the same original phrase: Deus ex machina. The phrase had a specific meaning related to plays, however both the artworks in question are using its literal translation: God from the machine.

Deus Ex, the video game series, has protagonists who undergo progressive cyborg augmentation. They gain god-like powers by adding machines to their bodies. The title leaves off the "Machine" but it's still there and implied.

Ex Machina, the movie, leaves out the Deus part of the phrase. I would argue that it's left ambiguous at the end just what emerged from the machine that is the main female character. God? Devil? Nothing more than a program executing its instructions? It could also be a play on the english meaing of "ex": formerly a machine, but no longer.

Ex machina is a Latin phrase which can be translated as "from the machine" or "by reason of the machine". If one starts from the second translation then "deus ex machina" can be taken to mean "god by reason of the machine".

Hmmm. Well, at the end of the movie it shows her in a city, as if to become part of society. It would be assumed that she will try to pass herself off as human. "Ex" means "used to be", or "formerly", as in ex-wife. She is now, at least socially, an ex-machine. So for me, "Ex-Machina" was the punch-line at the end of the movie; "Oh, That's what it means!"

In Greek "Deus ex machina" would translate as "O ??? ??????? ????", which is a term originating for ancient Greek dramatic poetry and especially from tragedy.

It meant that a God would be making his appearance before mortals, usually happening at the end of a play thus helping the writer ending his play, by providing a solution or resolving a rather difficult and dead-end situation

There are more than one theories why the term came to be:

1) An actor playing a God, would be lowered into the theater scene with the help of a wooden crane, it would appear as he was descending from the heavens.

2) Greeks Gods would descend to earth on flying chariots and they where trying to replicate this effect so the spectators could interpret the divine intervention.