My understanding is that aircraft use GPS systems for navigation (maybe not the only way to navigate). I would like to understand what would happen if the GPS system fails. Are there any other navigation systems to fall back on? What would happen if all of them fail? Could all the navigation systems fail?
It only takes 3 satellites for a GPS fix (4 if you want altitude). There are 31 operational satellites currently in orbit and at any given time you can most likely receive a signal from half a dozen of them at least. That being said the chances of them failing completely is almost 0%. But of course what if they did, well...
There is always the Russian version of GPS known as GLONASS so there is effectually an entire redundant global navigation system in space. While not all aircraft are equipped with GLONASS receivers it is of course becoming more common. Even the little tiny receiver I use for GA flights has it now. So there is that redundancy.
In controlled airspace you have ATC there to help you. They can effectively watch you on radar and tell you what to do which is basically what flying IFR is. For this you will need an operational altitude encoding transmitter so that ATC knows your altitude along with position. If you are flying VFR and require assistance you could call for Flight Following here in the US, load permitting. They can offer limited guidance but if you tell them you are lost they will generally help.
If there is no ATC you can hone in on VOR stations, or some of the older radio direction finders like ADF systems assuming your aircraft is equipped with such equipment. Some modern handheld radios can also hone in on VOR stations
If all that fails you have your good old magnetic compass and hopefully some way of keeping time. With close track of your airspeed, a clock, and a compass you can navigate pretty accurately even in the blind (although I don't recommend that).
No compass, no problem, you always have your ground reference points to get where you need to be. Basic VFR navigation which every pilot know how to do is learned this way. This will require you have a VFR map of the area which most GA planes will carry. Note the roads, power lines and other large things marked on the map that are easily identified from the sky.
Outdated/less popular systems: Inertial navigation systems are still used in plenty of places however they are not as accurate as other systems available in modern aircraft but they work by knowing how fast you have gone and in what direction you have accelerated to that speed. Thus position can be calculated. INS systems suffer from integration drift which can add up over time and make the margin of error pretty big.
There was once a time in history when you could get a navigators rating on your license. These people were trained in many times of navigation including celestial navigation. Older planes were even outfitted with a sextant for that very reason.
LORAN C was officially shut down here in the US/Canada in 2010 but there are still active chains elsewhere in the world. Some planes out there still have the units in them (one of the Cherokees I fly still has it sitting there). So depending on where you are you may be able to use it.
What Time Is It: The US GPS network is synced up via the USNO master clock and while it is unlikely that all the satellites will fail simultaneously or fall out of the sky, or be shut down. The master clock does provide a central point of failure for the system. While I am not up to date on the full function of the system I would imagine that a downed clock could introduce a margin of error into the system. Worse so an incorrect value fed from the master system may cause an erroneous reading.
The probability of the entire GPS system to fail is quite low. Individual satellites may fail, however that can be detected by the GPS receiver itself. Usually there are more satellites in view than minimally needed so the faulty satellite will be excluded from the position calculations.
If no GPS position can be calculated, for example because of a failure of the GPS receiver, the aircraft has a number of alternative navigation systems.
In no case GPS will be used as the sole means of navigation. There are always other navigation systems used in parallel.
Aircraft are equipped with VOR receivers, which uses signals from ground based navigation beacons to determine their direction. These VOR stations have a range of up to 200 NM. This isn't of much use of the oceans of course.
Inertial Navigation Systems calculate the position by keeping track of all accelerations and rotations of the aircraft. The accuracy degrades over time but it is good enough to find your way over the oceans. The good thing about INS is that it is totally independent of external systems.
If your GPS receiver (or the entire GPS system) fails, what you have left depends on what you have to begin with. Many GA aircraft that have been upgraded with a GPS receiver (and quite a few that came stock will full glass) still also have a good old navigation radio stack with at least one Omni Bearing Indicator:
This indicator works with a radionavigation system called VOR (VHF Omnidirectional Range). Ground stations at known locations broadcast a two-part signal; an omnidirectional radio beacon that identifies the station, and a sweeping, phase-shifting transmission of the same modulated signal that the nav radio compares with the omni signal. The amount of phase difference between the omni and sweeping signal is the same as the compass direction or radial from the station to the plane.
Pilots can use this system to fly "Victor routes" between major cities by selecting the VOR radio station they want to fly toward or away from, then dialing in the desired radial they want to approach along (which may be important to avoid obstacles or restricted/controlled airspace), then flying diagonally to that course to intercept the radial before turning toward or away from the station As they approach or pass one station, they can tune in to the next VOR station and radial indicated on the chart.
Using two radios, they can also triangulate their location based on the readings from two VOR stations; fly along one radial until you intercept another radial of a different station, and now you know you're at the intersection of those two radials on the map.
VOR (and other radionavigation methods like the military equivalent TACAN) are not quite as accurate as GPS, nor quite as reliable in all scenarios (many of these systems require line of sight which is more difficult in mountainous terrain or at very low altitude) but it's been the system that U.S. civil aviation has depended on since the 1960s, and if you know how to use it properly it works quite well.
A fail of the GPS system wouldn't cause any issue to properly trained pilots. Remember that aviation started in the early twentieth century! However if you have no visibility and all the navigation systems (or, more probably, just the pilots) fail, then you are in trouble and you can eventually crash. Just a couple of examples from the top of my head:
If you get completely lost at night in the middle of the Amazon forest you will probably run out of fuel and crash (see Varig Flight 254).
If you think that your navigation system is guiding the plane while it's accidentally off, you may end up on a nearby hill instead of the runway (see First Air Flight 6560 - note that even the GPS was working correctly but the pilots were confused).
In modern aircraft, this would happen (FMC intern uses these on priority basis 1. GPS 2. DME 3. VOR 4. IRS). In old aircraft their order remains same without FMS in place. https://en.wikipedia.org/wiki/Flight_management_system#Position_determination
There's a bunch of alternatives for the case that GPS fails. I guess that they would first resort to VOR-navigation ( http://www.pyrochta.ch/flash/start_vor.htm ). And if all else fails you'd still have a compass, a map and a clock for timing.
But I think that, if all communication failed as well, at some point some fighters would guide you to the next airport.
You will be never in a situation where you don't know anymore where you are and where you're going.