Funding is a general term. Funds = money, and if you want to do anything that costs money, you need funding. Funding just means any money that you receive. Graduate school costs money, and so they expect that you will need funding.
You can receive different types of funding, including a TA/RA or a fellowship. Graduate schools assume you will need this funding to pay your tuition, pay your health insurance, and to pay your living expenses. If you obtain "outside funding" in the form of a fellowship, the school will not have to fund you, but if you don't, the school will have to fund you.
The standard is for them to pay your tuition and health insurance expenses, and then for you to either teach or do research in exchange for a "stipend" which is the money you use to pay your living expenses. For example, Cornell pays my tuition and a basic health insurance plan, then I do research in a lab for an annual pre-tax income (stipend) of about $30,000, so all together I get about $60,000 per year in funding. If you obtain a nice fellowship, the fellowship will pay your tuition, health insurance, and stipend, in which case you wouldn't have to TA or do research unless you wanted to or unless it were a condition of your fellowship.