The ESP (all of the them) use normal serial communication (RS-232 on TTL-level), so you can use that to talk to it from an Arduino-board or you can use an ESP as a board on itself.
The ESP is even faster than an Arduino (80MHz opposed to 16MHz for the Arduino), though it has less I/O-pins than an Arduino.
If you want to use it as a "Dumb WiFi modem" for your Arduino, you can talk to it using AT-commands the standard firmware supports (just like telephone modems from the old days).
When you want to make full use of the ESP's potential, you can write your own firmware for it, with for instance the Arduino IDE.
After you've installed the ESP8266-library (using the boardmanager or manually) into the Arduino IDE, you can use the Arduino environment to compile and flash your ESP: just make sure it's connected to a serial port (via a USB -> TTL serial adapter) and you're good to go.
There are even ESP development board with an Arduino footprint.
There are lots of manuals and examples to be found all over this forum and the internet: just Google it.
So in short:
- You can use the ESP as a WiFi-module for an external board or controler (or even for a PC if you'd like)
or - You can use the ESP itself as a single-board computer (actually a SoC) with WiFi possibilities
I suggest, since they practically give them away (< $5), you first buy a development board that has everything on it (correct power supply, USB -> TTL serial convertor, flash- and reset-buttons, etc.) to get you started.
Have a Google for Wemos D1, NodeMCU, Witty board, etc.
If you have any questions, you're always welcome to ask them here.