Cersei stopped being Queen Regent when Joffrey married Margaery.
This is evidenced in the episode, around 39 minutes in, when Prince Oberyn Martell introduces his paramour, Ellaria Sand, to Cersei and Tywin.
He says: I don't believe you have met, Ellaria.
This is the Lord Hand Tywin Lannister and Cersei Lannister the Queen Regent.
I suppose it is former Queen Regent now.
This indicates that she is no longer the Queen Regent.
Of course, following the death of Joffrey, and the passing of the throne to Tommen, she once again continues to be Queen Regent.
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Yes, Cersei was (probably) still Queen Regent.
The "Regent" part of her title means she is ruling on behalf of a King who is underaged.
She would not stop being Queen Regent unless Joffrey became an adult, or someone else replaced her in the job.
This would seem to be in contradiction with Oberyn's remark that Cersei is the "former Queen Regent", mentioned in Moogle's answer.
There are a few possibilities: Under the laws of the Seven Kingdoms, Joffrey was considered to be an adult as soon as he was married, so the regency came to an end.
This seems unlikely -- if he isn't considered old enough to rule, then being married should make no difference.
Joffrey was still underage, and Cersei was still Queen Regent, but Oberyn either misunderstood her position, or deliberately pretended to misunderstand, and took the opportunity to insult Cersei.
[Edit:]
Joffrey had already come of age before
he was married.
So in principle, Cersei had not been Queen Regent for some time.
However, given Joffrey's youth and mental instability, and Cersei's greed for power and disregard for the letter of the law, she might have held onto her authority as Regent and perhaps even continued to use the title.
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tl;dr:
Cersei's position as regent ended when Joffrey turned 16, but exactly when that happened in the show isn't clear.
It certainly had nothing to do with Joffrey getting married.
In the novels, Cersei is still Queen Regent at this point in the story.
Joffrey won't be considered an adult until he is 16, and in the novels he's only 14 when he gets married and dies, at which point she is Tommen's regent.
In the show, the question is a lot more complex.
Most of the underage characters in the show are several years older than their book counterparts (largely because of Danaerys being only 12 when she marries Drogo and gets pregnant).
Thus, by this point in the show, Joffrey should already be 16, which means, he shouldn't have a regent, he is King in his own right.
I'm not exactly sure of the time scales in the show, but it possible that Cersei had no right to name herself regent in the first place.
Out of universe, I'd have to chalk this up to a continuity goof on the writer's part, stemming from their age advancement of Joffrey.
In universe, there's no real explanation for Cersei's title, or why Oberyn thinks she "lost" it, in any official capacity.
However, I can make a guess, based on the political maneuvering at King's Landing, that something like this is happening:
Cersei desperately wants to be Queen, so when her husband dies, she takes the opportunity to name herself regent for Joffrey, deserved or otherwise.
Given her connections both inside and outside King's Landing, no one argues with her.
Joffrey's behavior clearly indicates that he's not mentally mature enough to be King, regardless of his physical age, so when Cersei, and later Tywin, act on his behalf as Queen Regent and Hand to undo some of Joffrey's screw ups, people just listen to them.
Now, however, there is another powerful woman in Joffrey's life -- Margaery -- whom he is starting to listen to more than his mother.
This diminishes Cersei's control over Joffrey, which was probably never official to begin with.
As soon as Margaery marries Joffrey, she becomes Queen, and gains some official authority.
This, plus Joffrey's obvious support for her, basically pushes Cersei into the background.
Oberyn was basically making a dig at Cersei, letter
her know that he understands the dynamics going on in the King's court, and how much power she is losing there.
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The line of succession falls to the other recognized children of Robert and Cersei:
Prince Tommen and Princess Myrcella Baratheon.
Tommen is the younger, but I believe he gets precedence for being born with a Y chromosome.
Myrcella may not officially be in the line, I forget if/when George Martin explains the role gender plays in Westeros Succession.
I know it's different among other lands.
At any rate, Cersei is mother to them both, so she would still be the Queen Regent, and you can bet that she wants to remain in power.
However, as Shevliaskovic points out, the line of succession is muddied by the events immediately prior to your question.
I'm not sure they get around to answering the exact situation before the end of A Storm of Swords , other than to overtly indicate some tensions between Baratheon/Lannister and Highgarden.
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Cersei was not Queen Regent because the King's Wife, Margaery, was an adult (16 years old).
That means that she would be queen Regent.
The whole point of Cersei as Queen Regent was to rule the Kingdom (with Joffrey) until he comes of age.
Since he got married to a girl that was already of age, Cersei 'was no longer needed'.
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SPOILERS FOR SEASON 4 EPISODE 2! DO NOT READ AHEAD IF YOU HAVE NOT WATCHED THE EPISODE OR READ THE STORM OF SWORDS    After the royal wedding ceremony AND before Joffrey dropped dead (10 to 20 hours est.), was Cersei Lannister no longer the Queen Regent during this short period of time?
Game of Thrones: ex Queen Regent (season 4/Storm of Swords spoilers)
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In the novels, Cersei is still Queen Regent at this point in the story. She would not stop being Queen Regent unless Joffrey became an adult, or someone else replaced her in the job.  However, some claim that Cersei was not Queen Regent because the King's wife, Margaery, was an adult. Queen Regent is replaced by Margaery during this period because he married Margaery when she was of age at 16 years’ old. 
Queen Regent is replaced by Margaery during this period because he married Margaery when she was of age at 16 years’ old. 