As Emyr said, there's only one meaning of "horsepower". It is just a unit of power, like "watt" is a unit of power. Power is the rate of doing work or delivering energy, but if you don't already know that it's time to do some reading.
Motors and engines are rated under different conditions. That's what the "continuous" versus "peak" difference is about: the electric motor in the link can briefly put out 55 kW (74 hp), but it will soon overheat at that power; it can only sustain 25 kW (34 hp) continuously. Car engines are rated for what they can continuously sustain.
There's also that factor of speed. The charts with the linked motor show a typical characteristic for a high-voltage AC motor: it can produce its maximum torque from zero speed up to a rated speed (3,000 rpm for the smallest motor of this range, the APEV10), then it can produce the same maximum power at any speed from the rated speed to nearly the maximum speed. Other types of electric motor (such as the brushed DC "forklift" motors) have much less useful speed ranges. Engines produce their peak torque over some part of the middle of their speed range, and need to run faster than that to produce maximum power.
The extra speed flexibility of a suitable electric motor (which the right battery) means that the motor can produce maximum power over a wide road speed range without changing transmission ratios. Since modern automotive transmissions have 8 to 10 ratios and shift automatically, this isn' a performance problem for engines; however, people comparing poorly driven cars with engines and manual transmissions find that the speed flexibility with an electric motor appears to give them more impressive performance.
So, for an example, if an engine can produce at least half of it's rated power over a wide speed range, and it doesn't have a good transmission or a good driver, it might perform about as well as an electric car of the same weight with an electric motor rated at about half of the engine's rating.
For practical examples
the Chevrolet Bolt has an electric motor rated at 200 horsepower, and is about as quick as a modern engine-driven car of similar weight and perhaps slightly higher rated engine power - not advantage to the electric.
Tesla Model S Dual-motor cars are rated at several hundred horsepower, and are about as quick as gas engine cars rated at several hundred horsepower
So if you want the performance of a 250 horsepower gasoline engine, you need an electric motor which can produce almost 250 horsepower (at least peak, if not continuous). In the series of motors in the link, that's the APEV80-12, which needs 540 volts and weighs 260 kg. But you are not building a truck...