Funding during graduate study (phd, masters) in math in the US I have some questions and concerns in mind:


When somebody says a program is "funded", does that mean the money is conditional (whether he TAs or not) or unconditional? Or is it clarified in context?
Is there such a thing as entirely unconditional funding? If it's not tied to teaching, will it always be tied with progress (or something else)? 
Are there programs that will ask you to take out loans? In other words, do (most) schools generally have some sort of funding available? How does this depend on rankings/tier? Is there a specific ranking, say 50, below which you'll mostly likely need money through loans? Does it depend on type (masters or phd)?
For the sake of avoiding confusion, what are different sources? Fellowships, scholarships, the department, "TAships" (what is an exhaustive list?)


My personal concern is whether or not I can depend on being funded, even if I'm not a very strong student (or if I'm a student coming from a different discipline).
When somebody says a program is "funded", does that mean the money is conditional (whether he TAs or not) or unconditional? Or is it clarified in context?


The vast majority of funding is conditional on some amount of teaching, TAing and/or grading.  The exact amount varies from program to program and, often, within a given program, both from student to student and from year to year.  Various government-sponsored fellowships will cut out all teaching obligations for entire years, but there will be other years in which teaching is required.  Most students spend five or six years in a math PhD program in the US, and most prestigious fellowships don't fund all of that time: e.g. the NSF graduate fellowship provides three (very highly paid) full years of support.  You should also be aware that some teaching experience is desirable bordering on mandatory for future career success, so even a candidate who needs to do no teaching whatsoever (e.g. a foreigner supported by their own government) usually does at least one or two classes.


  Is there such a thing as entirely unconditional funding? If it's not tied to teaching, will it always be tied with progress (or something else)? 


NSF funding is not conditional on teaching (the above link makes this clear).  Yes, all funding is conditional on satisfactory progress.  It has to be, because continuation in a PhD program is conditional on satisfactory progress and funding is conditional on being in the program.  


  Are there programs that will ask you to take out loans? In other words, do (most) schools generally have some sort of funding available? How does this depend on rankings/tier? Is there a specific ranking, say 50, below which you'll mostly likely need money through loans? Does it depend on type (masters or phd)?


The majority of PhD programs in mathematics only want to admit students who have (in some technical sense, anyway) full funding, through a combination of stipends and teaching.  Students who want to pay their own way are generally admitted in "non-degree programs" or in ad hoc temporary situations.  I don't know of a PhD program that advises students to take out loans.  In fact, I would advise the contrary: you should not attend a math PhD program unless you have full funding.  What constitutes full funding may be higher at a top program (but also depending upon the location of the program, the wealth of the institution and other factors) but so far as I know, less prestigious programs are no less likely to fund their students.

Master's programs are different: in many cases, it is harder to get funding for a master's program, though it is certainly still possible.  In some cases there is an across the board raise from the master's program to the PhD program: e.g. the same TA duties done by a master's student may earn less than if done by a PhD student.  Still I would not advise any student to pay tuition to enroll in a math master's program in the US.  


  For the sake of avoiding confusion, what are different sources? Fellowships, scholarships, the department, "TAships" (what is an exhaustive list?)


There is external funding and then there is internal funding.  External funding comes from a variety of sources, especially the government, especially the NSF.  Internal funding comes from the department and/or the university.  It is usually divided into a stipend -- which is what you get paid just by being there -- and an instructorship, which is what you get paid for doing tutoring / grading / TAing / teaching.  For more precise information you should consult the individual department.


  My personal concern is whether or not I can depend on being funded, even if I'm not a very strong student (or if I'm a student coming from a different discipline).


The entire funding package should be given to you in detail when you are admitted.  It will not be a year-to-year deal: it will be guaranteed (conditional on satisfactory progress, which essentially means not getting kicked out of the program) for N years, where N is a number that will be given to you in advance and usually lies in [4,6].  Asking how often students get funded in years N+1, N+2, N+3 are good questions, by the way.