What It Is
When two pages answer a search about equally well, Google leans toward the one it has more reason to trust. The trade calls that trust authority. Google does not publish an authority score. The scores some SEO tools sell under names like Domain Authority are those tools' own estimates, not Google's. What Google does say is that it studies how pages link to each other, that it works to surface more authoritative pages, and that some of its signals apply to a whole site as well as to each page.
It is easy to mix up with brand equity, and the two are different. Brand equity is what customers believe about your name. Authority is what Google believes about your website. A beloved shop with a brand new site has plenty of the first and almost none of the second.
How It Works
Authority builds slowly. Links do most of the work: when a local paper, a supplier, or a neighborhood association links to a page, Google reads that as someone vouching for it. Time matters only because links take time to collect. A page that has stayed at the same web address for years, saying the same thing, has had years to gather them.
That is why authority belongs to web addresses, not to the business. The links point at one page on one domain, so when that page moves, the credit stays behind unless Google is told where it went. That is the job of a redirect, an instruction that sends anyone asking for the old address on to the new one. Google treats the two kinds very differently. A permanent redirect (a 301 or a 308) says the page has moved for good, so Google shows the new address in its results and moves the old page's signals over to it, links included. Its site move guide says permanent redirects "don't cause a loss in PageRank," the part of Google's ranking that weighs links. A temporary redirect (a 302 or a 307) says the move is not final, so Google keeps showing the old address and does not treat the new one as the page's real home.
Why It Matters for Being Found
Most of the authority a small business has is tied to its old web addresses, and a site move is where it gets lost. A new site on a new domain starts from nothing if the old pages simply stop working, and a redirect left on temporary tells Google the old address is still the one that counts. Done well, every old page redirects permanently to the new page that covers the same thing, never all of them to the home page, which Google warns against. Expect rankings to wobble for a few weeks or more while Google catches up. Then keep the old domain registered and redirecting for good. Google asks for at least a year and suggests keeping redirects indefinitely, because the links other sites gave you years ago keep pointing at the old address long after you have moved.