Yes, most electric watches will require a battery — this is where electricity will be stored. Some electric watches, however, can charge themselves using kinetic energy when you move your wrist, but unless your electric watch is automatic, you'll need batteries for it.
In the case of digital watches, the battery will usually be the CR1216 battery, but check the user manual for the exact model. For smart watches, though, you'll need to charge the device up just like your phone.