What do mathematics graduate schools look for in undergraduate students?
I am an (associate) professor of mathematics at the University of Georgia and was involved in graduate admissions for several years.  My answer is based on a perspective which is somewhat skewed towards pure mathematics: we have applied mathematics in our department as well, but it is one of seven research groups, so when looking at applications, to a first approximation we select for the same things among pure and applied applicants (and in fact not all applicants clearly identify themselves as one or the other, which is fine).  There are other departments that are more than half applied mathematics, and yet other departments (e.g. at Harvard) where applied math is a separate department entirely.  

The main things we look at are competitive  GRE scores (including the math subject exam), strong grades attained in strong courses, and very strong recommendation letters which convince us that the student would have a good chance of succeeding in our program.  These letters generally come from faculty who have taught the student multiple courses, including an advanced course and/or a reading course.  Getting one letter of recommendation from someone who directed undergraduate research is a popular choice and is fine, but it does not really make a big contribution to the application because most such letters sound exactly the same (no one is going to say "So and so did an REU with me, and let me tell you: we got nothing done over the whole summer.")  


  [S]chools actually place stronger emphasis on performance in 4 classes: a proofs/logic course, analysis, discrete math, and a linear algebra course (this was apparently has less emphasis than the first 3). 


I don't find this to be especially good advice.  For me, the key courses are 
(i) real analysis, (ii) abstract algebra, (iii) topology, especially the first two. These are in fact three of the four topics in which most of our PhD students take qualifying exams [some applied students take numerical analysis and/or probability].  Moreover, every graduate program in math I've ever heard of requires students to take exams in real analysis, abstract algebra or both.  Because these courses exist at both the advanced undergraduate and beginning graduate level, we really want students to have taken the undergraduate versions and have done well in them.  If they haven't, then they need to begin a graduate program taking these undergraduate courses, then make the transition to the graduate versions and then take and pass the quals, all within a couple of years.  That's hard.

Concerning your list:

(i) proofs/logic

Many students do not take a separate course on this.  In my undergraduate program we have a course on this but the strongest track of student avoid it by taking a kind of analysis class in which this material is taught on the fly.  Whether a student knows proofs and logic should be clear from the rest of their coursework and their recommendation letters.

(ii) analysis

Yes, this one is dead-on.  Let me say that a lot of courses are called "analysis".  We look carefully to see what the text is and what the topics were.  Something at the level of Rudin's Principles of Mathematical Analysis should be taken in order to be competitive.

(iii) discrete math

No, this is really not a critical course.  I can't think of many graduate courses that have such a course as a prerequisite.  Also the name "discrete math" is often used for a lower level course not even taught to majors.  But if you mean something like combinatorics, probability or graph theory: these are courses which most students enjoy, but they are not the courses we are looking to see if our applicants have taken.  This is not to say that the material from these courses is not important or useful in mathematics: it most certainly is, increasingly so across all parts of mathematics.  But it's not hard enough to learn to separate applicants like certain other courses.  I have never taken a course in graph theory, but some of my papers contain results about graphs: it's just not that hard to learn graph theory on the fly.  

(iv) linear algebra

Yes, you need to have taken such a course, just like you need to have taken a course in multivariable calculus.  But this is (at the latest) a sophomore-level course for most students who go on to graduate school.  A course (or more) in abstract algebra, which will in places build on the linear algebra, is much more critical.


  I was told that these classes are the bread and butter of most graduate math classes and are indicative of how well you can learn the more rigorous content. Is this true?


As above, I don't think the list is well-chosen to be among the most important courses.  Also "indicative of how well you can learn the more rigorous content" is a red flag to me: a student's graduate application should be replete with evidence of how well she has learned the more rigorous content, by successfully completing coursework containing this content!  

Who is giving you this advice?  Faculty in your department?  


  Do these schools care about extracurricular such as volunteering experience?


Not really.


  I know these aren't going to get you into Harvard per se, but do they slightly augment your profile? I know academia is primarily concerned with research so this may not be too relevant. 


Very slightly at best.  Just putting it on your CV is probably worth precisely nothing.  If you can describe your volunteering experiences in your personal statement in a way which makes them sound compelling and relevant to your graduate career then....well, then you have written a good personal statement, which could help you a bit.  If you did something really substantial which shows 
off organizational and administrative skills that most students lack, that would also help a bit...but not as much as the time you put into it.  Obviously there are other good reasons to do volunteer / community work that have nothing to do with an academic application, so you should certainly feel free to do so.  Just don't think of it in terms of improving your application.

Added: The OP asked for followup information from an applied perspective.  I can only give the perspective of a math department that admits some students who study applied mathematics.  In our department, the requirements for applied students are only slightly different from all other students.  In particular, see here for qualifying exam information, which shows that all students are required to take both real and complex analysis and (algebra or topology); students who do not want to take both algebra and topology can take probability (which by the way is very close to real analysis and quite challenging) or numerical analysis.  I believe that the vast majority of students in my PhD program specializing in applied mathematics have taken the abstract algebra qual.

My department has recently added on applied area of emphasis for its undergraduate math major.  In this track, abstract algebra is not requrired (though it is required for all other majors): see here for some sample programs.  On the other hand, as a student interested in graduate study you need to know that these major requirements are very intentionally designed to be as minimal as we can get away with.  We want to have more math majors, not less.  Students who want to take more classes are always free to do so.  Moreover we tell our students that for certain career paths they should take more classes.  Especially here we recommend further classes, including two semesters of algebra.  This advice is not specifically targeted at students who are interested in pure mathematics.  And by the way, there are many kinds of applied mathematics which you would be absolutely locked out of without an undergraduate background in algebra.  


  I do not think I will need abstract algebra directly, but I'm slightly ignorant.


It is very unlikely that any undergraduate math major can say with any assurance what kind of mathematics they will not need in their PhD studies or later career.  Making this kind of decision is a kind of negative investment in the future.  


  Some schools of interest do not have that content in the curricula. 


Please let me know what specific programs you are looking at.  That would be useful information to me and I could give you more targeted advice.  In particular, all of this advice is predicated under the assumption that you are applying for graduate study in a mathematics department in the United States.  If you are interested in an interdisciplinary program, or e.g. want to do mathematical biology from a biology department, then things will be very different.  If you are applying to a university which has separate programs in pure and applied math -- or has separate posted admissions requirements -- then things will be different again.  But for general admission to a math department in the US: not having abstract algebra will be a huge point against you, I believe.