Building a vocal booth is a lot like building a room inside of a room, though a vocal booth is intended to be acoustically altered to record vocal audio tracks without background murmur. One common misconception is that vocal booths are the same as isolation rooms. Vocal booths don't have to be sound proof rooms, and though many are designed with sound proofing in mind, this doesn't have to be the true objective. The main purpose of a vocal booth is to isolate the vocals so they're essentially the only thing that is being picked up by any microphones in the booth.
Because most vocal booths require quite a bit of construction, you'll likely need some wood for the process. If you have other materials available to you, they can be substituted. Using 2 by 4 pieces of wood to build the frame, you'll probably want to build your vocal booth in the corner of the room to conserve on materials. You'll want to plan on 6 inches of leeway inside of your booth when you begin the dampening process, and you'll likely want your room to have a 4- to 5-foot diameter.
Plasterboard is a great way to cut out low end noise. When applied to the inside of the room, the plasterboard will help eliminate bass noise on both sides of the booth. This can be especially helpful inside the booth, where you'll need to limit certain frequencies from resonating back into the microphone. One great way to do this is to build a bass trap on one of the walls to cancel out those low frequencies.
Rockwool is great at dampening noise. You should use thick Rockwool, at least 30mm thick, spaced at least 4 inches from your plasterboard walls. Rockwool can also be used to dampen the sound inside the room when hung a few inches away from the walls or hung on wooden spacers. Rockwool can also be used to line, fill and coat walls, the door and under the floor.
Acoustic foam is a special kind of foam that is designed to cut down on the liveliness of a sound space. Acoustic foam is typically pointed like egg crate bed foam, but it is much denser and does a better job of absorbing high end noise. You should use a large piece of acoustic foam to dampen the ceiling, where sound is likely to reverberate, and on each wall, where sound is carrying. Remember, you want the inside of your vocal booth to absorb most of the sound so it doesn't echo.