Terminal hacking is a significant skill in that gives you access to a wide range of narrative/combat options. Often, accessing the right terminal and choosing the proper commands can help you bypass huge enemy swarms or turn the tide of battle in your favor.
If you want to unlock sealed doors, understand life before the War, and access some of the biggest secrets in the game, you'll need to be a good hacker.
Hacking itself involves finding a terminal. Terminals are usually boxy and old-fashioned, fitting the 1950s aesthetic of the world, but they can also be circular or attached to the wall.
If it has a keyboard, it's probably a terminal. While many terminals have freely accessible information, important info is locked behind passwords.
What makes hacking in Fallout 4 so fun is that it is not instant access to a terminal but rather a mini-game that requires you to search through lines of code for the correct password.