Sura

Sure! In most graduate programs in the United States that fund their graduate students, there are 4 main sources of this money:
1. Fellowships/scholarships: These is money awarded to you without any actual expectation of work. 
2. Research assistantship: This is money paid to you in exchange for research work. In some fields (e.g. STEM ones like mine), this is research work towards your thesis/dissertation. In other fields, this is usually research work not related to your own thesis. For example, you might be paid an hourly wage to do a task like data entry for another person's research project.
3. Teaching assistantship: This is money paid to you in exchange for teaching work. In some fields, this is mostly just TAing: you grade papers, you run lab sessions or tutorial sesisons. In other fields, this may involve being an instructor of record and would have higher teaching responsibilities (i.e. preparing lectures or even course design).
4. Other assistantships: There isn't a good name for this because it varies, but this is other work on campus that isn't #2 or #3 above. For example, you might work in the registrar's office and do administrative tasks. Or, you might help students schedule appointments with the Writing Center. Not all schools offer this type of work for funding and it's usually only programs that cannot afford to fund you via the other three options that would result in you having to do this work. Thus, this is often considered a "last resort" because it's the type of work that will not help you advance your academic career.
These are just the most common ways that schools fund their students. There are other ways that might be a variant of the above. For example, one of my old graduate schools in Astronomy would hire/pay a graduate student to run the Outreach program at the school's Observatory at the same rate as a TA and the student would be doing this work instead of TA work. 
Finally, I notice that you are writing about these opportunities as if you will be able to simply choose to take on more work if you want more money. Unfortunately, this is rarely the case. In fact, these opportunities (even Type #4 above in places with little funding) are scarce commodities. That is, it's not like departments have unlimited TA and RA ships to hand out to any student that wants to work. Instead, they only have a limited pot of money to fund students and the way the funding is doled out is through these TA and RA work assignments. Usually, the amount of students they are able to accept is limited by money, so there is usually just barely enough TA and RA spots to give everyone who needs funding a job. In some rare cases, there may be one or two extra TA spots that someone might be able to choose to take for extra pay, but this is pretty rare. 
Therefore, I would not count on any opportunity to earn more than the stipend stated in your offer letter. These stipends are set according to how much resources the department has, i.e. how much money they have to pay RA and TAs. Usually, if grad students need to earn a little bit more, they may take a job on the side, e.g. a part time job, or a tutor (tutoring is especially popular in Physics and similar fields; one of our biggest client demographics are pre-med students!). However, as I said above, as an international student, you are limited by the laws on what you are allowed to do as work (tutoring is not allowed unless you are working for an on-campus group, but then you are still limited by # of hours).

I had a failed application cycle last year due to my own hubris. Many things snowballed with that, resulting in a pretty crummy year!
If I've learned anything from that mistake: Be happy. Do things that make you happy. Be in a place that makes you happy.

^ Absolutely. I am also often surprised by the "I'll settle for a less than ideal location for the next 5 years of my life because then I'll be able to get a better job" sentiment, which seems to forget that people will almost always have more options to choose from when it comes to graduate school than when it comes to the academic job market, even if they are the very top players. The chances that you will have to spend a few years in a less than ideal location for your first job (or three; assuming that you are lucky enough to get a job in the first place!) are incredibly high. Research fit that's good on paper means very little when you live in a place that makes you unhappy, trust me. 

There are a bunch of recent posts where people have given the kind of advice you're seeking, OP. Here are links to two of them. 
There's also a recent post where I argued for the importance of considering location.

Hi @Sura Definitely agree with the above. You also have a pretty solid list of secondary factors there, which though non-exhaustive, cover most of the important factors IMO. I'll also put down my thoughts and it would be interesting to get feedback from others.
I'd definitely pay attention to the funding package. Fellowships provide greater opportunity to explore and pick the perfect adviser. Even if you know whom you'd love to work with, and the person has agreed to take you, it's still possible that bad things might happen. For example, there could be clashes in personality, work-ethic, cultural issues etc. These are hard to predict without actually working with said person, and hence financial independence for at least the first year is a big bonus.
Some universities have heavy teaching obligations (1 sem or quarter every year); some others provide funding packages where you don't have to teach if you aren't interesting; and some roll out the red carpet. These are important considerations too. I got a 4 year Armstrong fellowship from Columbia which pays a LOT and gives me complete freedom with zero obligations. I was heavily leaning towards UW, but this has made me think again about Columbia.
I think "fixed" factors should be considered with higher priority. For example, you are annoyed with traffic and long commute - pretty easy to get another apartment. Need an additional $250, easy to get some grader or web admin position which aren't very time consuming. Unhappy with research area or adviser, much harder (without fellowship) to change. Unhappy with the climate or "culture" - impossible to change.

Hi all,
I must say that there are some fantastic answers from post-graduates here like that of @juilletmercredi's endless advice on spending time productively at grad school. I am sure there are many more enlightening posts on the same topic. It would be really helpful if people like her could give meaningful life advice for incoming grads at the stage of making decisions.
My dilemma is one that a majority of my peer group, I believe, is facing - research and the guides are two important factors upon which one chooses a grad school, but when one just can't make up her/his mind based on these 2 factors alone (like in my case where all options seem equally attractive and even my current guides pointed out that I could not go wrong with any decision), what other extraneous factors creep in to have an enjoyable 5-6 years at the grad school?
I do strongly believe, contrary to the popular opinion, that factors like the social life, weather and location too play a significant role in choosing a grad school. Your research is going to decide on your career, what you do apart from that will shape you as a person which is also imperative. In fact, I could also say that having a great life outside of your school would hold you in good stead through the difficult PhD times, right? Doesn't mental and physical well-being reinforce intellectualism? Also, for someone like myself coming from a tropical climate, weather is indeed a crucial factor. Not having been accustomed to freezing temperatures, wouldn't it take a toll on my health in the long run? That would definitely affect my research too, wouldn't it?
I am sure there are many more factors based on which one must make their decision, I simply can't think of anything more. It would be of great help to incoming grads like myself if experienced people like @juilletmercredi guide us in making an optimal choice for someone who does not have the mentality of "put-the-career-above-everything-else" and making a compromise on other things even if it means a dx improvement on your research.
Any answer would be appreciated. Thanks in advance!

@Unimpressed3D
Hi,
I had the exact same problem. Some places contacted me quite early. My guide suggested me to be professional in my approach. Tell them that you really like their program, ask a lot of questions about their group/funding/school in general. Remember that they have been doing this for a long time and they understand that you are looking forward to hearing from other places as well. In case they ask you to make a decision, you MUST let them know that you have applied to other places as well and are looking forward to hearing from them as well. You don't want to commit yourself prematurely to the place, however high it's ranked on your list. Let them know you need time. Most likely, they won't ask you about other places (they just wanted me to let them know once I was close to a decision - either acceptance or rejection), in which case you are fine. It's always good to sound diplomatic and at the same time excited. Do take the advice of professors in your school too as they would definitely guide you better.