2005 Ford Taurus. The car's engine runs hot, what should be checked?
Check the obvious things first. Any coolant leaking? My water pump failed, and was leaking from a little hole in the pump. Check all hoses, radiator, heater etc for wear/cracks/holes. Next the thermostat, it's a cheap part that often fails. (take it out, drop it in a cup of boiling water, if it opens it is working)
Is the fan working? Should be on when the car is very hot, (example sitting at the lights with no air flow) if not I'd try disconnecting it, and run wires to the battery. If that works, you know the fan is good, and the problem is a fuse, or the temp sensor, or worse.
Always use coolant not tap water, (except in emergency) as corrosion can form and damage your engine.
If it's really hot ?? it could be a head gasket, cracked or warped head. (check for steam coming from your exhaust) and coolant loss.
Get it fixed ASAP, because thermostats, hoses, and water pumps, and fuses and fans, are much cheaper and easier to replace and repair, than your engine.
Good luck.
PS, check your oil for water (grey not black or brown), and oil in the coolant, both signs of bad head gasket, or cracked /warped head.