It depends. The most common is that the car will just not run. From there it can be a lack of pressure, which will cause a variety of issues. These can be idle fluctuations, poor performance, lean O2 codes, etc. Sometimes there are other issues which will have the same symptoms. Lack of fuel pressure is the starting point for all fuel pump issues. Then you have to check for wiring issues, and internal fuel tank hose issues, all of which can have the same symptoms as a failing fuel pump.
I don't think you gets lots of warning when a fuel pump is going out. I just had mine die a few weeks ago. My car was running fine but I had about 95K on the fuel pump and I let the gas get pretty low in the tank because I wasn't paying attention. I pulled into a gas station and everything was still running fine. I cut my engine off to wait for the customer ahead of me to finish getting gas. When I tried to start it back up it would barely run. I put gas in it and was able to get it started and got a few hundred yards before it died again. After that I called a tow truck. Once I put a new fuel pump in all was well. I think letting my fuel get too low in the tank was the nail in the coffin for my fuel pump but like I said it was running just fine before it died.