Roughly ordered by how often we find them, with what each one means.
Thermostats and heating elements (electric). The most common electric failure by a wide margin. Upper element or thermostat gone: no hot water at all. Lower element gone: hot water that runs out fast. Both are inexpensive components and both are replaceable in a single visit. A high-limit cut-out that has tripped is related — and it trips for a reason, so repeated resetting without diagnosis is treating the alarm rather than the fire.
Thermocouples, pilot assemblies, and igniters (gas). The most common gas failure. A pilot that won't stay lit, or a unit that won't fire. Inexpensive parts, straightforward replacement, and the unit typically returns to full service.
The drain valve. A frequent leak point on ageing tanks, particularly plastic ones that have hardened. It looks like a leaking water heater and it isn't — it's a fitting, and it's replaceable.
The dip tube. The internal tube delivering cold water to the bottom of the tank. When it fails, cold enters at the top and mixes into the outgoing hot supply, so you get a fraction of the tank's capacity and hot water that runs out dramatically sooner. Cheap part, dramatic symptom, and it's very often mistaken for a tank at end of life.
The anode rod. A sacrificial rod that corrodes deliberately so the tank doesn't. This is maintenance rather than failure — but when it's exhausted the tank becomes the anode, and that's the start of the countdown to a leak. Inexpensive relative to a heater, and the single most effective thing you can do to extend one. Also the most commonly skipped item in the trade, because nobody sells a service around a part the customer has never heard of.
The temperature and pressure relief valve. The valve itself is inexpensive. What matters is why it's discharging — excessive temperature, excessive supply pressure, thermal expansion in a closed system, or a failed valve. Three of those four are conditions to correct rather than parts to swap. And it must never be capped, plugged, or restricted under any circumstances.
Sediment. Not a part, but the cause behind rumbling, popping, reduced capacity, longer recovery, and premature tank failure. Flushing removes it while it's loose. In a tank neglected for a decade, much of it won't come out — which is the argument for starting the habit early rather than when the noise begins.
The gas control valve. Here the arithmetic changes. It's a substantially more expensive component, and on an older unit it frequently costs enough that replacement becomes the sensible choice. This is the genuine pivot point between repair and replacement on a gas heater.
The tank itself. Not repairable. If water is coming from the tank body rather than a fitting, valve, or connection, the steel has failed and it doesn't improve. No patch, no sealant, no economical intervention. This is the one situation where "you need a new water heater" is simply the truth.