A smudge stick is a bundle of dried herbs, bound with thread, lit at one end and blown out so it smoulders. The most common is white sage. The practice of burning herbs for cleansing is close to universal; the specific word "smudging" is not.
Where the word comes from, and why it is worth knowing
"Smudging" refers to a specific set of ceremonial practices belonging to Indigenous peoples of North America. The commercial white sage bundle sold across Europe borrows that word for something considerably more casual. Plenty of people use it without a second thought, and we are not going to tell you that burning a herb bundle in your own home is wrong.
But it is worth knowing that the word carries a history, that some of the communities it belongs to have asked for more care with it, and that "herb bundle" describes the object equally well. That is the honest position: not a prohibition, just the information.
The white sage supply question is real
Unlike the palo santo panic, this one has substance. Salvia apiana grows in a limited range in southern California and northern Mexico, and demand has driven unlicensed wild harvesting on a scale that does concern conservationists. Cultivated white sage exists and is fine; wild-stripped white sage is the problem.
If that matters to you, garden sage, lavender, rosemary, cedar and mugwort all bundle and burn perfectly well, and several of them have longer European histories than white sage does.
Lighting one without setting off the alarm
Light the tip, let it catch for ten to twenty seconds, then blow the flame out so it glows. A bundle that is genuinely alight is producing far more smoke than you need and will not stop politely.
Keep a heatproof dish under it the whole time — a bowl, a shell, anything that will not mind an ember. Embers fall off bundles constantly and that is how carpets get burned.
Putting it out
Press the lit end into sand, ash or dry earth until it stops. Do not use water: a damp bundle will not relight properly and the inside stays wet for days, which encourages mould in the middle where you cannot see it. Never leave one smouldering in another room.
Ventilation is the part that is skipped
Open a window. Not because the smoke needs somewhere to go for ritual reasons, but because burning plant material in a closed room produces particulates, and a small room fills faster than people expect. If anyone in the house has asthma, this matters more than any of the above.