A trade secret is information used in a business which is protected, typically by physical measures and privacy agreements, from competitors and the public. Typical trade secrets include secret formulas for products (e.g. Coke or KFC seasoning blends), customer lists, pricing information, and proprietary credit scoring formulas.
Trade secrets are primarily protected by state rather than federal law. Theft of a trade secret, e.g., by spending spies to break security or encouraging someone to disclose information in violation of a privacy agreement, is typically punishable by civil and criminal penalties. Most common cases involve key employees who leave one business and attempt to seek employment with a competitor, sharing trade secret information to the deteriment of the former employer.
Trade secrets are unlimited in duration. They last as long as the business can keep the secret. But, they provide no protection against reverse engineering of the trade secret (e.g. buying KFC and taking it to a lab for analysis), independent discovery of the protected information (e.g. making your own customer list from scratch), or from people who use lawful means to obtain the trade secret after it has been unlawfully disclosed (e.g. disclosure in a court document which the trade secret owner's lawyers fail to have sealed).
Patent law exists, in part, to provide an alternative way to protect ideas to trade secrets when the limitations described above are important (e.g. an idea that can easily be reverse engineered or independently discovered once you have the general idea). In contrast, trade secrets can protect many ideas which wouldn't be entitled to patent protection (like a mathematical formula, a recipe or a customer list) which never the less have business value.
Obviously, trade secrets rarely overlap with trademarks (except in a brief pre-release period for a new trademark).
Generally speaking, trade secrets are minimally invasive, in a direct sense, to the public domain, a matter of grave concern in other areas of intellectual property. But, an intellectual property regime which strongly favors trade secrets also means that many valuable ideas may never make it into the public sphere. In contrast, when an idea is patented, it must be made public and will eventually become a part of the public domain.