Types of Law: Common Law v. Civil Law
The 'common law' is made up of legislation and judges' decisions, which are treated as precedents. The United States (US) has a common law system. When it took/gained its independence its courts were applying the same case law precedents that applied in Britain. It has continued to use those precedents although, of course, it has developed many thousands more of its own. The US will, of course, prefer its own precedent decisions. Modern English precedent decisions can be, and often are, cited in American courts and American precedents can and are cited in British courts. The same applies in the courts of Canada, Australia, New Zealand, Ireland and other common law countries. The common origin and heritage, of these countries, means that the basic structure and principles of many of our laws are quite similar. The details differ but there is much that is common. Of course English precedent decisions are not binding on the courts of another independent country. Similarly their precedent decisions are not binding on us.
Another country's legal precedents are not binding in this country. Recall that a 'precedent' is a case about the meaning of a law, for example whether someone can be guilty of murder if they were reckless. That is not the same as enforcing the factual decisions of the courts of another country. Imagine, for example, that another country's court has decided that a child should live with its mother, rather than its father. Perhaps the mother is in that country but the father is keeping the child in this country. This is not a problem about 'precedent' decisions. Rather it will depend upon a statute. If that country, and this, both have signed an agreement (in this example it would be the Hague Convention), to respect each others' courts' decisions then that is what will happen. It is important to appreciate that 'case law' and 'precedents' refer to interpretations of the law - and not to decisions of fact (e.g. verdicts). Similarly when someone is extradited from one country to another to face trial there.
Two great systems, common law and civil law
Are you Scottish? If so are you fuming because your country has not been mentioned? Sorry if you are - fuming that is - but you create problems. Scottish legal history is more closely connected with continental Europe than it is with England and Wales. It has a distinctively different legal system, different courts, titles and terminology. In many respects in resembles a Civil Law country as it values treatises on the law much more than in common law countries.