When you ask a court to do something for you, such as grant you a divorce, you must give a reason for your request when you file your documents. "Grounds" is the legal term for a reason that's acceptable under a particular state's statutes. Not all states recognize the same grounds, however. A legally acceptable reason in one jurisdiction may not be recognized in another.
In 2010, New York became the last state to recognize no-fault divorce, so this option is now available in all jurisdictions. In some states, such as California, you can only file for divorce on no-fault grounds. In others, you have a choice between no-fault or fault-based grounds. Additionally, the exact criteria for each state's no-fault grounds can differ. In California, for example, you can simply tell the court that your marriage isn't working – you have "irreconcilable differences"; in Maryland, you and your spouse must live separately for a year.
In states that also recognize fault-based divorces, you usually have a wide array of allegations to choose from. They generally include things like adultery, cruelty and desertion. Tennessee is among the more creative states – it recognizes a total of 15 different grounds, including "attempt to kill one's spouse." If you cite fault-based grounds in your divorce papers, most states require that you present solid evidence, proving that your spouse really did what you're accusing him of. He has the right to raise defenses against your allegations, and if he's successful, the court can dismiss your divorce action.