Contract means usually fixed duration, whereas contract to hire means it begins as a contract, but with the potential/desire to hire the person full time.
Contract - Independent means they hire you directly, as a 1099 which is the designation of the tax code of you you report your income.
Contract - W2 means they hire you directly, as W2. W2's are typically full time employees with benefits, but could be a contractor
Contract - Corp to Corp is means they want to hire a contractor through another corporation, as opposed to independent. This is usually because they want legal protection for liabilities and other issues. An independent contractor usually has limited funds, where as a contract definately carries some insurance, has deep pockets, can provide a replacement if something happens to the original contractor, etc.
Mark Herschberg, author of The Career Toolkit
to me "Corp to Corp" just means they want to be dealing with a corporation. This gives the client the greatest legal protection. You can have a one person corporation and still be contracting "corp to corp". If you're not incorporated, though, you'd need to find a headhunter to work through.
"Independent" means they will pay on 1099, which means you get to file a schedule C and pay both the employer and employee halves of social security. This is kind of risky for the client because the IRS and many state governments may claim that you're actually an employee rather than a contractor, which gives them liability (such as for any taxes you might not pay). Headhunters are more willing to pay on 1099, because the fact that you're not working directly on their premises, but instead on client premises, gives them a bit of legal protection.
"W-2" means you're legally an employee. Usually only headhunters will do this, since clients would pretty much have to treat you as an employee if you were on W-2.
"Contract to hire" means they consider the contract period basically a trial, and will want to hire you at employee salaries if you work out. People who prefer contracting sometimes avoid these, because it means they aren't going to want to extend the contract if you don't want to become an employee.