When a child custody case requires a judge to make a decision regarding custody, then the judge's decision can be appealed. However, most child custody cases are resolved via agreements between parents that get approved by the court.
Settlement agreements are not appealable, though if the judge made an error in approving the agreement, that may be appealable. However, appeals are typically only used when a party believes a judge made an error (legal or otherwise) in reaching their decision. Depending on each state's civil court procedures, the time for filing an appeal will vary, and can range from a matter of a week or two up to about a month. When a parent is not happy with a child custody agreement or order, at anytime after the order becomes final, they can petition the court for a modification.
Appeal Versus Modification
Filing an appeal is a much more involved process than requesting a modification of a previous child custody order. Generally, courts will allow parents to petition or request the court to modify a previous order if there are certain changed circumstances, or after a period of time has elapsed. For example, if a parent who previously did not have a bedroom for a child to sleep in moves into a home that can accommodate overnight visits, they may petition the court to modify the order to allow such visits.
While a modification request can be filed at any time, an appeal can only be filed within a limited time period after the judge enters their final judgment on the matter. In addition to an appeal, some states allow motions for reconsideration to be filed, which is basically a request for a judge to review the judgment they just issued and points the judge towards certain facts they may not have considered.
As a practical matter, filing an appeal should be carefully considered with the help of an attorney. While occasionally an appellate court reverses a judgment or modifies an order, most frequently appellate courts will only confirm or reject whether the lower court made an error, if they do anything at all. If an error is found, the appellate court will order the lower court to review their judgment in lieu of the error, or to retry the case.