You can start and own a US company without a visa, and without even coming to the US. Managing an LLC from outside the US is allowed, but may be not be permitted within the US without a valid work visa. Being a director, as well as a shareholder, of a US corporation is allowed without any kind of visa, but being an officer and performing your duties within the US is generally not allowed. Working for your corporation or LLC within the US without a valid work visa is not allowed.
If you are planning to come to the US to start a business, you will need to have one of the various types of US business visas. Without a visa, you can be the owner or manager of a US company, but you cannot legally work before. If you do work for your own company without the proper visa, you can be deported without right of return, and your company can be fined for hiring an illegal alien.

TYPES OF US BUSINESS VISAS

B1 Visa
Description: Short term business visa. Not allowed to sign contracts or perform work for hire;
Length: Temporary, up to 6 months;
Limitations: Can negotiate but cannot sign contracts.

B2 Visa
Description: Short term tourist visa;
Length: Temporary, up to 6 months;
Limitations: Can negotiate but cannot sign contract.

E-1 Visa
Description: Treaty trader visa. Good for setting up a business that will trade with the visa-holder’s home country. Must be with a country with an appropriate treaty with the US;
Length: Temporary, can be renewed until business is no longer in operation;
Limitations: Is only valid as long as there is a business. Children under 21 cannot work, and lose their status once they turn 21.

E-2 Visa
Description: Treaty investor visa. Allows a citizen of a treaty country to come to the US to start a business;
Length: Temporary, can be renewed until business is no longer in operation;
Limitations: Is only valid as long as there is a business. Children under 21 cannot work, and lose their status once they turn 21.

EB-5 Visa
Description: Investor Green Card program. Requires an investment of $900k to $1.8M, must hire at least 10 US residents within 2 years and survive more than 5 years;
Length: Leads to permanent residence after probation period.

L-1 Visa
Description: Intercompany transfer visa. Allows holder to transfer from foreign company to a US company subject to restrictions;
Length: One year, with up to 3 extensions;
Limitations: Employee must have worked for foreign, related company for more than one year in the last 3 years.

H-1B Visa
Description: Specialized labor visa;
Length: Three year, extendable;
Limitations: Annual quota restricts number of visas available. Cannot be used for self-employment.

O-1 Visa
Description: Extraordinary ability visa;
Length: Up to three years, extendable;
Limitations: Must be able to document extraordinary ability.

TN Visa
Description: NAFTA temporary work visa;
Length: Up to three years, extendable;
Limitations: Cannot be used for self-employment.