Mark has pretty much explained the general application process, I'll give a bit of information on the finance/economics PhDs area...
US:
-Although quite a lot of science PhDs require GREs, most finance programs require the GMAT (some accept GREs, but GMAT is preferred).
-As far as funding goes, similar to what Mark has said: a combination of fellowship, TA, RA should provide enough financial support.
-Getting into the top ivy league programs is a bit of a hit and miss and do sometimes largely depend on luck. Finance programs are extremely competitive, to maximise your chances you should (at minimum) satisfy the following:
1. Strong academic achievements and very clear and direct research interest(s). You should have close to perfect academic grades and thorough research experience throughout your undergraduate studies. (The key here is research).
2. Another very important component is recommendation letters. A bit of 'politics' is often present here if you have "connections", goes a long way...
3. At least in finance, getting a few publications under your belt will REALLY strengthen your application (aim for top tier journals, JBF/JFE/JFQA etc), especially if you can co author with someone from the institution that you're applying to (preferably the person that you would want to supervise you for your PhD).
4. Close to perfect score on the GMAT.
5. Try to get accepted and attend as much prestigious finance conferences as possible, in the US (AFA/MFA) or EU (EFA/MFS).
Even given the above preqs you may still get rejected from top institutions simply to due to the sheer number of top tier applicants. Sometimes it comes down to luck whether you get the offer or not.