1. Do a Google search for the entity name and see what come up.
2. Do a trademark search at United States Patent and Trademark Office.
3. Do a domain search at domain registrar. (e.g. GoDaddy, Register.com, etc.)
If a domain is not available doesn't necessary mean you can't use that brand name. There may be some creative ways to get around domain name issue.
4. Go to Secretary of State website to search if someone is already incorporated under name.
Search for "California Secretary of State" on google. When you arrive at the Secretary of State website of your state, search for "business name search" within the site, then do your business entity research there.
5. Go to County clerk's office to see if someone is using a fictitous business name.
The best way to reach a website of your county clerk's office is by doing a Google search on, for example, "Alameda county clerk office." Once you reach the website, look for fictitous business name section by typing "business name search" into the search box. Search fictitous business name to find if anyone is already doing business under the name you're interested in.
After your brand name research and decide to go with it, it is not yours unless you own and trademark it. Anyone doing brand research can and should do the above easy ways. But if you're getting serious about the legal availability, you may want a trademark attorney to help you.