Branding irons for home use in cooking are a novelty / vanity item.
They are miniature branding irons for burning brand marks into food that is being grilled (aka barbequed in the UK.) The rod and brand parts are made of iron; the handle is wooden. The brand mark will be about 4 cm (1 ½ inches) wide.
Some makes, when you place your order for them, allow you to specify two or three initials for the monogram.
Some instead of initials are like three-side arrowheads. One side says R for “rare”, one “M” for medium, and the third side “W” for well-done. These are slightly more practical than the cooks’s initials, in that you can use it to mark the meat for various people’s preferences and keep track of it on a crowded grill.
Some come in presentation / storage boxes.
To use one, you heat it first in the fire, like a regular branding iron, and then “brand” away.