Armored vests are a somewhat common method of protecting one's body from lethal gunfire and sharp attacks (i.e. stabs, cuts, bites), as well as offering minor protection against explosions and the ever-present threat of clubs, maces, and muffalo. Armored vests can be purchased from traders, manufactured, or stripped from Outlanders and Pirates. Armored vests are typically encountered after the third or fourth raid, depending on difficulty and wealth (this assumes Rough and Crashlanded scenario, so somewhat 'standard' wealth).
Armored vests can be manufactured at a machining table, once machining has been researched. An armor vest requires 120 steel, 2 components, and 334 seconds of work (20,000 ticks) to be created, assuming baseline global work speed and no enhancements or hindrances to consciousness, sight, or manipulation.
Conclusion and Comparison
Armored vests offer slightly inferior protection, and vastly inferior coverage compared to Power Armor, but is also significantly cheaper to produce and purchase. Additionally: Power Armor can't be found on enemies, though elite Pirates may still have Power Armor Helmets. Armor Vests also only occupy the middle layer - meaning it can't be worn with tribalwear; whereas Power Armor occupies both the middle and shell layers, meaning that overgarments such as Dusters may be worn over the top.
Most, if not all, of your colonists should have an armored vest by the mid-game.
The following table lists how much protection an armor vest provides at each quality level.