The first step is figuring out whether the black water comes from the hot tap, the cold tap, or both. Run each one separately and watch. That one observation narrows the cause a lot, because hot-only problems live inside your water heater, while cold or both-sides problems point to your supply line, your pipes, or the city main. Don't skip it. It decides everything that follows.
If only the hot water is discolored, your water heater is almost certainly the source. Every tank heater has a sacrificial anode rod, usually magnesium or aluminum, built to corrode in place of the steel tank. As it does its job, it can release dark particles and, near the end of its life, black or murky water, especially once hard-water sediment has piled up on the tank floor. That's the anode protecting your tank exactly the way it's supposed to.
Here's the chemistry in plain terms. The anode rod is a more reactive metal than the steel tank, so corrosion attacks the rod first and leaves the tank wall alone. Once the rod is mostly consumed, it can no longer protect the tank, and corrosion shifts to the steel itself. The black tint is largely harmless mineral and metal sediment, but it's a clear signal that the anode rod is due for inspection and likely replacement.
If the black water shows up on the cold side or on both, look toward your supply. Naturally occurring manganese in groundwater can oxidize into black particles, particularly after the water sits. Old galvanized steel pipe corrodes from the inside and sheds dark, rusty debris as it ages, common in pre-1970s homes. And recent municipal main repairs or hydrant flushing can stir up sediment that lands in your lines for a day or two before clearing.
For a water-heater source, try flushing the tank. Shut off the cold inlet and the gas or power, connect a hose to the drain valve, and run it to a safe outdoor location until the water clears. If it keeps coming back black, the anode rod needs to be checked and replaced. For city-main events, run a cold tap for several minutes and the water usually clears on its own. If it doesn't clear within a day, call your water provider.
Call a plumber when the discoloration sticks around, keeps coming back, comes with a metallic taste or visible flakes, or when you've got old galvanized piping. Persistent black water from corroding galvanized pipe isn't a cosmetic issue. It means the pipe walls are deteriorating from the inside, and that line is headed toward leaks and reduced flow. At that point, repiping the affected runs is the durable fix, not flushing it again and again.