If a foreign court enters a judgment against someone by default that judgment will probably be enforceable in the country where the judgment was entered (or in the case of the U.S. in the state where it was entered with easy domestication to another U.S. state) against assets in that country. So, if you own real estate, or have bank accounts, or work, or do business is the forum state, you probably can't afford to ignore such a lawsuit.
Many U.S. judgments (especially in "tort" matters) are not automatically enforceable abroad, mostly because foreign countries are concerned about the non-economic and punitive damages available in U.S. courts in such lawsuits. Many foreign judgments (for example in defamation cases as noted in another answer to your question) are not enforceable in the U.S. (similarly, most judgments in Saudi Arabian courts are not enforceable due to a lack of due process in the court system there).
A foreign judgment in a contract case or in an arbitration forum agreed to be contract, is much more likely to be enforceable in another country, although what is necessary to "domesticate" a foreign judgment varies a fair amount. Some countries require what amounts to proof from scratch of the entire case almost, but may allow evidence from the prior case such as transcripts to be admitted, while other countries, particularly when a recognition of foreign judgments act applies, will routinely rubber stamp certain kinds of foreign judgments.
Many U.S. states have adopted the Uniform Recognition of Foreign Money Judgments Act, either in its original 1962 version or with 2005 amendments. The official summary of this act is as follows:
The first step towards enforcement is recognition of the foreign
country judgment. The recognition occurs in a state court when an
appropriate action is filed for the purpose. If the judgment meets
the statutory standards, the state court will recognize it. It then
may be enforced as if it is a judgment of another state of the United
States. Enforcement may then proceed, which means the judgment
creditor may proceed against the property of the judgment debtor to
satisfy the judgment amount.
First, it must be shown that the judgment is conclusive, final and
enforceable in the country of origin. Certain money judgments are
excluded, such as judgments on taxes, fines or criminal-like penalties
and judgments relating to domestic relations. Domestic relations
judgments are enforced under other statutes, already existing in every
state. A foreign-country judgment must not be recognized if it comes
from a court system that is not impartial or that dishonors due
process, or there is no personal jurisdiction over the defendant or
over the subject matter of the litigation. There are a number of
grounds that may make a U.S. court deny recognition, i.e., the
defendant did not receive notice of the proceeding or the claim is
repugnant to American public policy. A final, conclusive judgment
enforceable in the country of origin, if it is not excluded for one of
the enumerated reasons, must be recognized and enforced. The 1962 Act
and the 2005 Act generally operate the same.
The primary differences between the 1962 and the 2005 Uniform Acts are
The 2005 Act makes it clear that a judgment entitled to full faith and credit under the U.S. Constitution is not enforceable under this
Act. This clarifies the relationship between the Foreign-Country
Money Judgments Act and the Enforcement of Foreign Judgments Act.
Recognition by a court is a different procedure than enforcement of a
sister state judgment from within the United States.
The 2005 Act expressly provides that a party seeking recognition of a foreign judgment has the burden to prove that the judgment is
subject to the Uniform Act. Burden of proof was not addressed in the
Conversely, the 2005 Act imposes the burden of proof for establishing a specific ground for non-recognition upon the party
raising it. Again, burden of proof is not addressed in the 1962 Act.
The 2005 Act addresses the specific procedure for seeking enforcement. If recognition is sought as an original matter, the
party seeking recognition must file an action in the court to obtain
recognition. If recognition is sought in a pending action, it may be
filed as a counter-claim, cross-claim or affirmative defense in the
pending action. The 1962 Act does not address the procedure to obtain
recognition at all, leaving that to other state law.
The 2005 Act provides a statute of limitations on enforcement of a foreign-country judgment. If the judgment cannot be enforced any
longer in the country of origin, it may not be enforced in a court of
an enacting state. If there is no limitation on enforcement in the
country of origin, the judgment becomes unenforceable in an enacting
state after 15 years from the time the judgment is effective in the
country of origin.