First a word of congratulations. I applaud your effort to learn of this important topic and to do so coming froma background that would have discouraged this.
Secondly, a word of caution. Treat everything your read sceptically. Don't believe something simply because someone in authority says so. On the other hand if several authorities tell you the same thing and they provide the evidence to support that, then provisionally accept it as factual. (Everything in science is always provisional, it's just that some things are more provisional than others.)
As to your questions: blue eyes are both a trait and a mutation. They are a trait, in that their presence, or absence, is controlled by our genetic makeup.
If we inherit the genes for blue eyes from both parents then we will have blue eyes. If we inherit the gene for blue eyes from only one parent and for brown eyes from the other we will have brown eyes, since the gene for brown eyes is a dominant gene and for blue eyes a recessive gene. (It is actually more complicated than that: you can find a slightly more detailed, but still simple, explanation here:
http://www.brighthub.com/science/genetics/articles/22176.aspx).
But blue eyes, or rather the gene for blue eyes is also a mutation. It occured, probably a number of times in some of our ancestors, then spread through the population.
You ask why would it occur? Well most mutations are either neutral or harmful. They occur by chance, not as a means of improving the fitness of the organism. By chance a small percentage of mutations will offer some advantage to the organism.
I don't know the details for the blue eye gene, but here are some thoughts. Where did blue eyed people evolve? In the far north, where - trust me on this one - there is precious little sun, so that the disadvantage of the gene may have been quite minor, or even absent. Moreover, there may be some advantage confederred by blue eyes. Certainly in some societies the blue eyed person will be favoured over the brown eyed as a potential mate - that's sexual selection an important compliment to natural selection.
Also, the gene that controls the eye colour may be controlling other things too and these may confer advantages. The specifics in this area lay outside of my expertise.
Have you looked at wikipedia articles such as this one on genetics:
http://en.wikipedia.org/wiki/Genetics.
Don't try to absorb it all in one sitting, but reading a number of articles like that will allow you to become familiar with the language. There are many excellent books on evolution. I'm not a great fan of Richard Dawkins, but two of his works The Ancestor's Tale and The Greatest Show on Earth are both good. Also, as a direct repsonse to creationist arguments Why Evolution is True by Jerry Coyne may be a good choice for you.
Edit: I see I took so long to complete this that several others have already made the same points. My apologies for the repetition.