TAs, RAs, and almost all fellowships will cover the cost of your tuition and your stipend. The amount of stipend that you receive will depend on the specific fellowship you have, grant that your RA falls under, or teaching obligations. Most fields determine their grad students' funding situations on a semesterly basis. If you have a TA or RA, then the graduate field is required by the university to provide you with a letter informing you of the source and quantity of your funding. If you do not get this information, you should ask your advisor and/or your Graduate Field Advisor.
TAs last for one semester. RAs are usually only committed for one semester at a time as well. Fellowships may be for a single semester, or an entire year, though most often they cover tuition for the fall and spring and your stipend for nine months. If you want to stay at Cornell during the summer, there are many TA and teaching positions available and faculty can fund students on RAs over the summer as well.
Some funding sources will cover things besides your tuition and stipend. Many engineering graduate students get their student activity fee paid for them. Students who are ``fully funded" (most graduate students who receive tuition and a stipend are considered ``fully funded") will also have their mandatory health insurance premiums covered by Cornell (see Section 4.6.5 for more information on health insurance requirements at Cornell). Some fellowships, and some grants that RAs are under, can cover the cost of traveling to conferences or purchasing equipment. You should check if these things or anything else are covered for you.
About a month before the start of each semester, you may receive a bursar bill with the total amount of your tuition listed as due. Do not panic! If you have funding, you do not have to pay this (though if you have other charges that you are responsible for on the bill, you should pay those). The bursar office sometimes bills for tuition before the process of paying your tuition via a fellowship, RA, or TA has begun. This bill will also have your student activity fee listed on it.
It is quite possible (and perhaps even likely) that your tuition will not be paid before registration begins. You cannot register unless your tuition is paid. However, it is very important that you are registered since this maintains your student status, allowing you to use university resources and the health center, and also maintains international students' visa status. If your tuition is not paid, when you try to register you will be told to go to the registrar's office in Day Hall (C5). Once there, the registrar's office will tell you that the bursar's office has a hold on your account and will send you there. The bursar's office can release the hold on your account -- you need to tell them how you are supposed to be funded and the name of your department's contact person (usually the GFA). They will send you back to the registrar's office, and they will let you register.
When your next bursar bill arrives, it should show that your tuition has been paid. If it does not, then contact your GFA or funding agency and let them know about the problem. You should also check at this point that your student activity fee has been paid, if appropriate. You will probably have accrued some finance charges as well at this point. Any charges due to late payment of your tuition can be removed; ask your field's GFA to take care of this for you.