A business has a legal name and may trade under a different one.
The legal name
For a company, the name on its formation documents. For a sole proprietor, the owner's own name.
This is the name that belongs on contracts, invoices, insurance policies and bank accounts. A contract in a trading name that is not registered can be difficult to enforce.
Trading under another name
Registering a name to trade under, commonly called a DBA or a fictitious name filing, is what permits a business to operate publicly under something other than its legal name.
Requirements are set at state or county level. Where it is required and not done, the consequences range from fines to being unable to enforce a contract or open an account in that name.
It is also frequently what a bank requires before accepting payments made out to the trading name — see business credit.
What registration does not do
Grant exclusive rights to the name. Name registration is an administrative filing; it records that a business is using the name and does not stop anyone else from using it.
Exclusive rights come from trademark, which is a separate process at state or federal level with its own requirements. A business that has built a reputation on a name and never protected it can find it used by a competitor with no straightforward remedy.
Checking before committing
Whether the name is available as an entity name in the state, whether it conflicts with an existing trademark, and whether the matching domain and profiles are available.
Discovering a conflict after signage, vehicles and marketing have been produced is expensive and avoidable.
Consistency
The same name, address and phone across every listing matters for being found in local search — see Google Business Profile.