The turibulum or incense burner was used in household worship throughout Roman history. The turibulum is used both to create sacred scents pleasing to the gods, and also to change things from solid form into an ethereal form by consuming them with fire.
The turibulum holds hot coals, and powdered or resin incense is put on them to give off smoke. The coals were also used to burn small offerings such as bits food or flowers and other sacred plants.
In the ancient Roman world, the turibulum was made from a variety of materials depending on a person's needs or monetary status. The form varied as well.
To add a turibulum to your home lararium, you will need a non-burnable container, (clay, stone or metal), and fill it with some sort of non-burnable substance for the coals to rest on so that they won't make the turibulum too hot to hold or leave on the surface of an altar. A simple pottery bowl filled with an insulating substance such as sand (so that the coals won't overheat and crack the bowl) will work fine, as can a metal or stoneware vessel. The turibulum may be decorated or plain. Incense burners are of course commercially available in religious shops, etc.
The material for the inside of the incense burner should be both non-burnable, and also something that doesn't conduct heat. Sand is perfect. Clay based granular "kitty litter" will work as well. Dug up earth won't work well unless it is very, very dry, as anything organic in it tends to be burned by the charcoal and give off a smell.
In the ancient world, the coals for the turibulum were wood charcoal. Today it is easy enough to buy "incense burner charcoal." This can be purchased at many different stores including church supply stores, religious shops, new age shops, and of course on-line. Outdoor "charcoal briquettes" for your backyard grill should not be used at all in any ritual and definitely not indoors as they give off poison gasses that can be very dangerous if inhaled inside a closed space.