An ABN, or Australian Business Number, is an 11-digit number issued by the Australian Business Register to anyone carrying on a business in Australia. It is how the ATO, other businesses and government agencies tell your business apart from every other one. A sole trader has a single ABN for all of their business activity, whatever trading names they use, while a company, partnership or trust has its own. Holding an ABN is a separate thing from being registered for GST: plenty of small trades have one without the other, and the public ABN Lookup shows which is which for any number you search.
In practice
Homeowners rarely ask for your ABN. Businesses almost always do, and before they pay you rather than after. If a business pays a supplier who has not quoted an ABN, it may have to withhold tax from the payment at the top rate and send it to the ATO, so accounts teams hold the payment until the number turns up. Businesses in building and construction also report the payments they make to contractors to the ATO each year, matched to each contractor's ABN, which is another reason a builder wants yours on file before the first claim.
Take a tiler who finishes a bathroom for a small builder and invoices $3,600 without an ABN on the invoice. The builder's bookkeeper sends it back. A week goes by, the corrected invoice goes in, and it joins the next payment run a fortnight later. The work was done on time and the money arrived three weeks late because of eleven digits.
The fix is to put your ABN in the footer of your invoice template once, along with your business name as registered. If you also quote commercial clients, add it to the note on the quote, because their supplier set-up form will ask for it and the quote is often the first document they file. Check your own entry on ABN Lookup now and then as well: if your business details have changed, the register should say so before a client's system does.
For anything about whether you need an ABN, or what to do when a payer has withheld tax, check with the ATO.