It says that Training and Competence is needed.
The competence of a mage defines if they can use a magic or not.
this manga excerpt somehow explains this.
In fairy Tail Volume 28 chapter 233,
"Those who are atuned to the flow of magic can process and use any magic in existence."
Fairy Tail's Plot entry in Wikipedia says that:
The fictional world of Earth-land is populated by wizards who coalesce into guilds to hone their magical abilities and apply them to paid job requests.
However this might be due to the status of Wizards in the Land, as we can also see townspeople who can't or maybe doesn't use magic
According to FairyTail Wikia: http://fairytail.wikia.com/wiki/Mage
Mages (Madōshi), also known as Wizards, are individuals who can wield Magic of any kind.
Mages are wielders of Magic, using their Magic Power to cast spells and employ magical objects. They comprise only 10% of the living population of Earth Land, with the remaining 90% being either incapable or unwilling to use Magic.
Also, There is a note in the end of Fairytail's Wikipedia Entry:
In the references section, it is stated that, https://en.wikipedia.org/wiki/Fairy_Tail#References
According to the Fairy Tail Volume 2 Del Rey edition Translation Notes, General Notes, Wizard: So this translation has taken that as its inspiration and translated the word madôshi as "wizard". But madôshi's meaning is similar to certain Japanese words that have been borrowed by the English language, such as judo (the soft way) and kendo (the way of the sword). Madô is the way of magic, and madôshi are those who follow the way of magic. So although the word "wizard" is used in the original dialogue, a Japanese reader would be likely to think not of traditional Western wizards such as Merlin or Gandalf, but of martial artists.
Anyone Can learn Martial arts, but those who continue to learn and train can practically use it. There are also people who doens't want to.
In this answer to the question "What dictates who can use what magic in Fairy Tail?"
I think to be able to use a type of magic, one has to learn it. There are no requirements as to who can learn which magic.
In one particular episode, Mirajane teaches Natsu, Lucy and Happy the transformation magic and Natsu was able to transform into Lucy (even though that was not perfect, but that shows that you need to practice a particular type of magic).
And as have been mentioned in other answers that many characters are able to use more than one kind of magic.
And also Learning and WillPower
Magic has to be learned. When Erza was still a child, she was able to learn magic without actually practicing it. If you have knowledge of how to use a certain magic, you can use it. I guess it may depend on the user's willpower, or it may actually be inside them all along. By the way, there are a few characters who can use more than one type of magic like Ultear, Jellal, Zero/Brain, and Hades.
So to sum it up, one person might be able to learn Magic thru extensive training and practice of a certain particular kind of magic, has knowledge and understanding of it, Competence or Either he chooses to learn Magic or Not even if he/she is capable of it.