Most small businesses do not need special permits. However, many do. To find out if you need one or more, consult local and state authorities having jurisdiction over your area of business (see list of agencies, below.) People in your trade or industry can often tell you.
While not technically a permit, every business should be registered. A corporation is registered with a state agency via papers of incorporation and a sole proprietorship and partnership register with a local county courthouse by way of an “assumed name certificate.” Contact your local SBDC to find out how to register your business name in Texas.