If a BIA appeal is dismissed or denied, the facts of the case will determine the next move. But making the choice on what to do next is often tricky and may be influenced by how aggressively the government intends to pursue removal.
When the BIA dismisses an appeal of an Immigration Judge's decision to order a person removed and to not grant any relief from removal, the BIA's decision makes the removal order "administratively final." This begins something called the "removal period," a 90 day period in which the government is obligated to try to deport the immigrant. INA 241(a)(1)(A).
But in the 30 days following the decision of the BIA, the immigrant can "petition" a Circuit Court of Appeals to review the Board's decision. These two deadlines often cause tension - while it may seem to the immigrant that they have 30 whole days to appeal, Immigration is not always going to wait 30 days to arrest the immigrant and begin trying to deport them. Sometimes a Circuit Court will issue a stay of removal, but not always. Even if the court issues a stay of removal, DHS will not always release the immigrant from detention while the appeal is pending. Because of this, it is important to move quickly after receiving the BIA's decision.
While most dismissed appeals can be taken to the Circuit Court, that doesn't mean they all should, and you should consult with an attorney to discuss whether you have a good case to take up on appeal. Just because you are allowed to file something doesn't mean you always should, and if there is absolutely no chance of success before the court, you may be wasting your money and time. On the other hand, taking your case to the Circuit Court makes a lot of sense in many cases, and you should not rule it out. Often it seems as though the BIA did not consider one particular, important detail, or failed to account for the big picture.
If you do not file a "petition for review" in the Circuit Court within 30 days, your right to an appeal will expire. The only way to have the BIA consider your case after that will be to file a Motion to Reopen explaining why your case deserves another look. There are very strict limits on the type of Motion to Reopen that can be filed, the kind of evidence that must be attached, and the time limit for filing them. If you think reopening is warranted in your case, you should contact an attorney to discuss your options.