Important factors for hiring a young mathematician
I don't have anything essentially divergent to say from the other answers, but since you inquired about mathematics specifically and I am a mathematician who has been (and currently am) involved in postdoctoral and tenure track hiring, I thought it might be useful for me to weigh in as well.  Lacking true inspiration, let me just comment on your criteria.
(Let me also assume that we are talking about jobs at a &quot;research university&quot;, as it seems you are.)

Research experience (research articles, research talks, expository articles, perhaps books).

Mathematicians are judged on a combination of research promise and research success: as you get older, one looks increasingly for evidence that the former has been converted into the latter.  However, for every kind of research job in mathematics, they are hiring you on the basis of the work they expect you to do in the future.  So, for instance, if you have already solved a major problem, you can coast on that for a time but after a while people want to hear what you are working on now.  Research talks are probably the best way of exploring the dichotomy between past and future research: as such, they are very important whenever they exist, which is almost always on the tenure track job market and in a small (but perhaps increasing?) minority of postdoctoral jobs.
Expository articles generally do not count towards the research component of your application.  (If someone is counting papers, then if you have 8 research papers and 1 expository paper then people will probably say you have 9 papers altogether, so it counts a little bit.  But if pressed, its value could contract considerably: e.g. if there are worries that a candidate has too few papers, than a paper viewed as expository will probably not allay this worry.) Strategically it is probably best to advertise expository articles as having some teaching / mentoring / service component, if at all possible.
With regard to writing books: one of my most distinguished colleagues, Dino Lorenzini, wrote an excellent and rather successful book near the beginning of his career.  He now tells anyone who will listen that junior faculty should not write books.  Of course sometimes the heart wants what it wants, but from a strategic perspective I think this is eminently sound, and I say this as someone who may turn around and write some books now that I am solidly into my mid-career.

Teaching experience (tutoring, undergraduate courses, graduate courses, etc).

Successful teaching experiences are indeed valued for a research job.  At most research universities teaching is still a main component of one's job and, especially, of one's promotion and tenure packages.  Most research departments are looking most of the time for thoroughly solid teachers rather than especially brilliant or innovative ones.  Your teaching dossier should convey most of all that the department who hires you will never have to think about your teaching in a negative way.
Some graduate students do not get to do instruction at all (as opposed to TA work: grading, leading problem sessions, and so forth).  There is a big difference between TA work and instruction, and as a hirer I am very wary of candidates who attempt to convince me that they will be a successful instructor based only on TA experience.  I would strongly advise every math graduate student to be the instructor of record for at least one successful undergraduate course (where &quot;successful&quot; means you can get a strong teaching letter out of the experience).
Teaching experience at the graduate level is almost unheard of for graduate students and is far from guaranteed in postdocs.  Even within my own department, some of my colleagues feel strongly that postdocs should teach topics graduate courses in their areas of interest, whereas other colleagues feel that it is the job of the tenure track faculty to teach these courses.  I lean more towards the latter, and I don't feel that teaching graduate courses is an important part of a tenure track job application: I would be equally impressed or more with other kinds of interactions with the graduate program, e.g. participating in or organizing seminars, reading theses, and so forth.
I'm having a hard time thinking about how tutoring experience could play a significant role.  If you have founded the Khan Academy or the Euclid Lab, you would probably get some attention / consideration for this.  Much less and your employers are unlikely to care.

Academic background (grades, the university of graduation, current affiliation, etc).

No research university that I know of asks for grades or transcripts for candidates for faculty positions.  Of course your pedigree -- i.e., where you did your undergraduate and graduate work, and your current institution -- is of some importance, but not that much importance.  Anyway, what's done is done here: presumably you took what opportunities you could to attend / work at better, rather than worse, institutions!

Reference (who is writing reference letter and how he is writing it).

Or she!  When you are applying for your first postdoc, your adviser's letter may well be the most important part of the application: most graduates, even very strong ones, cannot speak about their research accomplishments and near future goals as convincingly as their advisers can.  Later on your adviser's letter gets less important, but you probably get more letters overall, and they are always a key part of the application.

Personal (natural) features (nationality, race, language, etc).

In the United States it is illegal to discriminate on the basis of race.  Nationality is probably a no-go as well.  Language issues are important: if English is not your native language, employers will (or at any rate, should) look carefully at your skills.

Social and family status (connections, marriage, etc).

Professional connections can be important; e.g. they come out in recommendation letters and in your academic pedigree.  It is illegal for employers to inquire about your marital status, sexual orientation, or whether you have children.  It is not illegal for you to bring these things up, and if you have a &quot;two-body problem&quot; -- i.e., a partner who is also an academic -- than you should do so at some point, and that brings an extra layer of complication to the process.  But if a candidate is not looking for an academic job for her partner, I honestly don't care at all whether she is married, gay, celibate, and so forth.  I can assure you that in most American departments any such talk about these matters in the context of a hiring discussion would be rapidly quelled.