Why amps alone don't tell you watts
Watts measure power, the rate energy is used. Amps measure current, the rate charge flows. The two are related but not interchangeable: the same 15 amps flowing through a 120V US outlet and a 230V European outlet delivers very different power (1,800W versus 3,450W), because power also depends on the voltage pushing that current through the circuit. This is Watt's Law, P = V x I, and it holds for any DC circuit or purely resistive AC load.
That is why a search for "amps to watts" cannot have a single fixed answer the way a length or temperature conversion does (Wikipedia's "Ohm's law" article). A calculator that only asks for amps and returns a watt figure is silently assuming a voltage, usually a standard household value, on your behalf.
DC, AC single-phase, and AC three-phase
DC circuits and purely resistive AC loads (space heaters, incandescent bulbs) use the plain formula P = V x I. Other AC loads, especially anything with a motor, coil, or transformer, draw current that is slightly out of phase with the voltage; power factor (PF) is the fraction of that V x I product that actually does useful work, so real power becomes P = V x I x PF. PF is always 1 for DC (there is no phase to be out of) and ranges from about 0.8 to 0.95 for typical motors, dropping lower for poorly designed or lightly loaded equipment (Wikipedia's "Power factor" article).
Three-phase power (common in industrial and commercial buildings) adds a second multiplier on top of PF, because three-phase delivers power over three conductors carrying current 120 degrees out of phase with each other. If the voltage figure you have is measured line-to-line (between two of the three phase conductors), the formula is P = sqrt(3) x V x I x PF. If it is measured line-to-neutral (between one phase conductor and the neutral), the formula is P = 3 x V x I x PF instead (Wikipedia's "Three-phase electric power" article). Using the wrong one of these two formulas for the voltage reading you actually have is an easy mistake to make, since both look almost identical on the page.
Worked example: a US household circuit
A hair dryer or space heater is rated 15A on a US 120V circuit, PF ≈ 1 for a resistive heating element. Watts = 120 x 15 x 1 = 1,800W. This is why a standard 15A US household circuit breaker is generally treated as good for up to about 1,800W of continuous resistive load (in practice, the NEC 80% continuous-load rule keeps sustained draw closer to 1,440W, but the breaker itself trips at 15A regardless of what it is powering).
Worked example: a three-phase industrial motor
A motor is fed by a 400V line-to-line three-phase supply (the standard European industrial three-phase voltage), drawing 20A at a power factor of 0.9. Watts = sqrt(3) x 400 x 20 x 0.9 = 1.7320508 x 7,200 = 12,470.77W, or about 12.47kW. Using the line-to-neutral formula (x3 instead of x sqrt(3)) on the same 400V figure by mistake would overestimate the power by a factor of sqrt(3), about 73% too high, which is exactly the kind of mix-up the circuit-type and voltage-type selectors above are meant to prevent.