Some boxes have slots along the walls, you are meant to cut your board to fit between them. Then just slide the board in.
A rather extreme number of slots, usually they are a cm or so apart.
Or you might mount the board on standoffs with bolts.
Many project cases come with plastic posts and screws to use in mounting PCBs inside.
This project box has both slots along the walls, and plastic posts on the bottom:
Here you can see the built-in plastic posts being used:
The person making this project used standoffs in the box, as his PCB didn't align with the posts:
You can even buy project boxes with soldered PCBs already made for them:
Sometimes parts are affixed to the board so that when things like potentiometers and jacks are screwed into the case, they are all at the proper height and all LEDs and displays are in the proper place and height. In that case, the parts are the board mount. It can be a bit tricky to get that correct.
Something similar can be done using short wires between the things mounted in the front panel and the board. Not nearly as critical, and you are simply relying on the very short wires to make up for any inaccuracies but be short and stiff enough to hold the board.
Obviously, not good for anything heavy on the board, like a large heat sink.
Sometimes, boards are held in by parts being mounted on a heatsink, and the heatsink is bolted down. Again, nothing heavy on the board like large electrolytic capacitors or large toroids, or you should also be using standoffs and mounting the board properly.
Having the board hanging by stiff wires or component leads isn't good if this is something exposed to a lot of vibration. In cases like that, for instance automotive use, use only stranded flexible wire and mount the board on standoffs, possibly even using rubber standoffs.
I have found that it is a good idea to use a bit of ribbon cable between LCD displays and the main board. Otherwise, normal knocking around can flex the LCD board from the weight and torque of the main board pulling on its connector, and there is only a conductive silicone ribbon between the LCD glass itself and the LCD driver board. Flexing that can make it unreliable.