If you want to run the server part local, yes, you need something to run the server on, for example, but not limited to, a Raspberry Pi (which is by far the cheapest solution to run the server). The Arduino can be anything really, as @Costas a Wemos might be better because it’s easier to manage, program and is way more stable then Arduino in combination with an ESP for example.
@bhaa @Lichtsignaal is the man to ask regarding local servers but to answer your question no an Arduino with a Pi is not generally sufficient for WiFi control of your home appliances.
You also need to add WiFi (or Ethernet but again not recommended for most users based on the relatively high cost and inconvenience compared with an “ESP” type module) capabilities to your Arduino.