What is the meaning of the term "motivic"? I'm looking for a simpler explanation of the idea behind motives, and a quick overview of applications.
One particular question is the meaning of the term "motivic" (which I've seen multiple times applied to different nouns, i.e. "motivic periods", "a [certain theory has a] motivic core", etc)
The Wikipedia article is unfortunately too unclear and convoluted for me to properly understand it. Introductory references will be highly appreciated.
 A: Take whatever I write with a lot of care, I do not guarantee completeness or correctness, and I welcome any corrections.
The issue is that motives have been around for quite some time (I would guess roughly 60 years now) and there is not even a complete definition that satisfies everybody. But there are a lot of conjectures on magic properties that they should have, and quite some evidence that research is indeed on the right track. Motives have been introduced by Grothendieck in the '60s because people realized that a "universal cohomology theory" for algebraic varieties could help to solve the standard conjectures about algebraic cycles (which are still open). The idea was that motives should contain the "essential core" of varieties and provide such a theory, but there was no satisfying definition of them. From then, various approaches have been studied. I will mention two, which are not necessarily the most relevant. Nori defined a category that could be the right one, but it is still not proved whether it satisfies all the required properties. Other people (notably Voevodsky) found a category which should be the derived category of the category of motives, if this exists. Even without the actual category of motives, one can do quite some things with derived categories, so they are actually useful. In an ideal world, it will be proved that the two approaches are just different points of view, and for some very special cases this has actually been proved ($1-$motives, for instance).
Motivic periods are, roughly speaking, the complex numbers that one obtains "integrating on motives". Some motives correspond to varieties and one just gets the periods obtained from integrating algebraic differential forms on that variety, but working with (whatever category) of motives is more general and sometimes easier.
Depending on your background, you might find this series of 4 lectures interesting. You will not learn everything about motives, but Brown is a great mathematician who masters both the mathematical aspects and the applications of motivic periods to physics, so his talks are really interesting in any case. As Dietrich Burde pointed out in a comment, some of his articles could also be interesting for you.
I don't think that there is a satisfying definition of what "motivic" means in full generality :)
The book "Une introduction aux motifs" by Yves André is also a very good reference, but it is probably not available in English.
Here is a related question on MathOverflow which might give you some more ideas.
In any case, keep in mind that motives are quite an advanced topic, so an easy introduction might not exist. If you find one, let me know, I would be interested as well ;)
