Contract law is the law that governs contracts. A contract can take several forms - it can be written on a piece of paper (or napkin, even) or it can be just a verbal agreement if it satisfies the Statute of Frauds.
A contract is basically an agreement between two or more people which creates an obligation to do, or not do, something. The agreement creates a legal relationship of rights and duties. If the agreement is broken, then the law provides certain remedies.
For a contract to be legally enforceable, not only do all the parties to the contract have to get something in return, but they must also suffer a detriment. In other words, one cannot contract to give someone else $500 unless the other person gives up something in return.
Contract law covers the legal implications of a contract. For instance, contract law determines what is and is not consideration, whether a contract was actually intended, if the parties making the contract were legally competent, whether there was fraud or duress involved, or how a contract is terminated.
Certain contracts are not legally valid, for instance, if the contract goes against public policy (such as a contract to satisfy a gambling debt). In most instances, people who are not of the age of majority are not allowed to make contracts.
When writing a contract, it is important to remember a central principle to contract law - terms of a contract will be construed against the drafter of the contract. Thus, any ambiguities or uncertainties will be resolved against the writer. For instance, if someone drafts a contract where there is uncertainty as to whether the drafter has to pay $500 or $800 for a good or service, the contract will be interpreted against the drafter - i.e. he or she will have to pay the $800 amount.