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Welcome to FT, fowler35!
At the outset, please use more descriptive thread titles, so other FTers can know exactly what your thread is about. As you might imagine, we get a lot of "newbie questions" here on FlyerTalk.
In most, if not all cases, membership in a frequent-flyer program can only be in an individual's name. Some airlines may also have a different program for rewarding a business for travel by the business' employees, but that is typically separate from that airline's FFP.
"Starwood" is a hotel chain that includes Sheraton, Westin, etc. It offers a frequent-guest program, Starwood Preferred Guest, and SPG points can be redeemed for free hotel stays, or transferred to a number of airline FFPs. Membership in SPG is separate and apart from membership in the FFP of any of its partner airlines.
If, on the other hand, you meant "Star Alliance" and not "Starwood," only airlines -- not individuals -- can join Star Alliance. If you belong to the FFP of an airline that is a Star Alliance member, you can redeem your frequent-flyer miles in that FFP for travel on any Star Alliance carrier. And if you fly on a paid ticket on another Star Alliance carrier, you can -- depending on the fare and airline -- credit miles for that flight to your FFP by providing your FFP number to the Star Alliance carrier on which you are actually flying.
Depending on how many miles you expect to fly in a year, what fares you are paying, and what benefits you are hoping to derive, it will sometimes make more sense to credit your flights not to the FFP of the airline you primarily fly, but rather to a different FFP that can provide greater benefits for your particular travel patterns.
Credit-card issuers issue both personal and business credit cards to individuals. An individual can apply for either type of card in his or her own name, and miles earned from either type of card will be credited to the cardholder's FFP. There may be exceptions, but as far as I know, a credit card opened in a business' name does not earn frequent-flyer miles for anyone, although cardholders of American Express corporate cards might have the option of earning points in the Amex Membership Rewards program upon payment of a separate annual fee.