Is an algebraic structure a mathematical structure? I am trying to get a sense of the relationships between various abstract math concepts. I keep seeing mathematical structures and algebraic structures mentioned and explained, but I never see them mentioned together. Any reason for that?
Based on what I've read, I am guessing a mathematical structure is the most abstract, while an algebraic structure is slightly more concrete version of a mathematical structure?
The definition of a mathematical structure as defined on wikipedia is briefly:

In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology).1 Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.

While an algebraic structure is defined as:

In mathematics, more specifically in abstract algebra and universal algebra, an algebraic structure consists of a set A (called the underlying set, carrier set or domain), a collection of operations on A of finite arity (typically binary operations), and a finite set of identities, known as axioms, that these operations must satisfy. Some algebraic structures also involve another set (called the scalar set).

Both definitions are to me somewhat circular in that understanding either concept require a lot of prior math knowledge. I am basically a programmer with basic calculus and linear algebra knowledge. I can derivation and integration as well as dot products, cross products and matrix multiplication. It is all very hands on. I don't know much about how mathematicians look at this stuff abstractly and categorize things.
I am not looking to pursue mathematics but get some handle on these kinds of concepts that pop up when I try to read about vectors, matrices and tensors.
 A: Yes, an algebraic structure is just a particular type of mathematical structure, one which designates operations performed with a set.
You can also have a topological structure on a set. That is, the set has subsets designated as "open" sets in a topology on the set. 
Or the same set could have both, an algebraic and topological structure, both of which are compatible in nice ways.
There are many other types of structures you can put on a set, each of which studies something slightly different.
For example, if you designate elements as "points" and some subsets as "lines" you can make an 'incidence structure' on the space. Or if you designate a partial order, you get a partial order structure, and so on.
I would say that "mathematical structure" is a bit of a catchall in the way that "quadrilateral" is. It's a "set endowed with features", which is pretty open ended.  Something like "algebraic" structure is much more specific and tells you what it is endowed with.
A: your question is a natural one, but you may be worrying unnecessarily.
whilst you are absorbing @rschwieb's answer and the useful points made in the comments below your question - particularly the contribution by @nickD - it may help to devote a little thought to the distinction between formal language and meta-language. 
example: in contemporary mathematics the term "group" belongs to the formal language, whereas "structure" belongs to the meta-language.
a term like "structure" is useful mainly because its meaning cannot be pinned down to a precise, inflexible definition. 
what is the difference between structure and system? what is the connection between terms like set, family, collection, class, ensemble? how do the following relate: function, functional, map, mapping, transform, transformation, operation, morphism? it seems that situational and semantic factors determine which of a cluster of near-synonyms is most appropriate in any particular context or sub-discipline.
contrariwise, to become a part of mathematics, a term must, at least in principle, always be used in a way which is completely well-defined.  
since mathematics is all about precision in reasoning the role of the meta-language is not always appreciated. metalanguage is useful to mathematicians, but its study is not necessarily of interest since this study belongs to psycholinguistics rather than to mathematics per se (though it has relevance to the teaching of math). 
some version of these considerations probably explains the frequent apologies encountered in mathematical writing for "abuse of terminology". this common practice might strike a non-mathematician as rather odd, but in fact it is a healthy sign that the logical conscience, which raises math above the level of the more pragmatic and worldly sciences, can never be entirely stifled. 
clearly it would be challenging to attempt a formal definition of "abuse of terminology". if strictly interpreted, this description might extend to virtually the whole of mathematics. in practice an experienced practitioner knows how to distinguish a useful abuse of terminology from an error, or from a bad choice of terminology which can lead to confusion and place additional obstacles in the way of comprehension. again, such 'knowledge' cannot be formalized. it is a knowing-how rather than a knowing-that
