Place one — upstream of the heater. Everything feeding it. What sits here: the gas supply valve, the breaker or electrical supply, the cold water inlet valve, and your supply pressure. What it looks like: no hot water at all, a unit that appears completely dead, or a heater that has stopped working with no warning and no noise. Worth checking first, because it's free: a tripped breaker, a gas valve someone closed, or a cold inlet valve that was shut off during other work. These are not rare, and they're the cheapest possible outcome. Also here: supply pressure that's too high, which stresses the tank and the relief valve continuously and is measurable in five minutes with a gauge on an outside tap.
Place two — the controls. What tells the heater to work. What sits here: thermostats and heating elements on an electric unit; thermocouples, pilot assemblies, igniters, and the gas control valve on a gas one; plus the high-limit cut-out on both. What it looks like: no hot water, water that's warm but never hot, a pilot that won't stay lit, or a unit that heats and then stops. What it usually costs: this is the good category. Thermostats, elements, thermocouples, and igniters are inexpensive components and generally worth replacing at almost any point in a unit's life. The one exception is the gas control valve, which is substantially more expensive and, on an older tank, frequently tips the arithmetic toward replacement.
Place three — the vessel. The tank and what's inside it. What sits here: the steel tank, the anode rod, the dip tube, accumulated sediment, and the temperature and pressure relief valve. What it looks like: rumbling or popping, hot water that runs out far sooner than it used to, rusty hot water, a relief valve that discharges, or water appearing underneath. The distinction that matters most in this whole category: water from a fitting, valve, connection, or the drain valve is a repair. Water from the tank body is not. A corroded tank can't be patched or sealed, and it doesn't improve. Two things people don't know are here: the dip tube, whose failure lets cold water mix straight into the outgoing hot supply and produces a dramatic loss of capacity from a cheap part — and the anode rod, a sacrificial rod that corrodes so the tank doesn't. When it's exhausted, the tank becomes the anode, and that's the start of the countdown. It's replaceable, inexpensive, and the most commonly skipped item in the trade.
Place four — downstream. Everything between the heater and you. What sits here: the hot water pipework, mixing valves, thermostatic and anti-scald devices, faucet cartridges, and the sheer length of run to the fixture. What it looks like: and this is the one that fools people — lukewarm water everywhere, hot water that takes forever to arrive, or one fixture that's cool while others are hot. The specific fault worth knowing: crossover. A worn cartridge in a single-lever faucet or a failed mixing valve can let cold water feed backward into the hot supply. The result is a whole property of lukewarm water and a water heater that's working perfectly. How to check it in two minutes: shut the cold supply to the heater and open a hot tap. If water keeps flowing, cold is entering the hot side somewhere in your plumbing — and no work on the heater will fix that.