For Apple Watch Series 1, the full range of features is available when Apple Watch and the paired iPhone are connected via Bluetooth and your iPhone has access to both cellular service and the Internet (via either cellular data or Wi-Fi).
You don't necessarily need to have your iPhone on or immediately around you (within arm's reach, for example) for the devices to establish and maintain their Bluetooth connection. Normal Bluetooth range is around 33 feet / 10 meters (this will vary in practice due to wireless interference).
Whenever your watch can't connect to the iPhone via Bluetooth, it will attempt, as a fallback, to connect to a known Wi-Fi network - this being a compatible 2.4GHz Wi-Fi network (some exceptions apply) that your iPhone has connected to before whilst your Apple Watch was connected to the iPhone via Bluetooth.
When instead connected to Wi-Fi, the watch also still offers a number of features, including receiving and sending messages via iMessage. If your carrier offers Wi-Fi Calling and it is supported on your plan and enabled on your iPhone, you can also use Apple Watch to make calls and (subject to carrier support) to send and receive SMS / text messages.
If your Apple Watch is unable to connect either via Bluetooth or Wi-Fi, it again still offers some - although fewer - features.
Features that are available in each case are detailed here:
How to use your Apple Watch without your iPhone nearby - Apple Support
For the best performance from your iPhone and Apple Watch, including the most power-efficient communication between them, Apple recommends keeping both Bluetooth and Wi-Fi permanently enabled on your iPhone. More information: