Appellate Court Personal Jurisdiction

Usually, as to Texas state class C misdemeanors or municipal ordinance violations, personal jurisdiction in a trial court is obtained over a person by service of a summons, though in rare cases, it can be gained by the arrest of the defendant.
In higher level Texas crimes, personal jurisdiction is normally  invoked by the arrest of the defendant.
In civil matters, normally, one or more plaintiffs give notice to one or more defendants by service of citation and the petition or complaint is served with the citation. Except for special appearances-- procedurally tricky filings which allege that a court does not have jurisdiction over a matter-- even an unserved party can subject itself to jurisdiction by merely filing a document in the court's file.
In justice of the peace courts, a timely request with a bond or a pauper's oath will get one a de novo trial in a higher court. The same is true of municipal courts that are not courts of record-- that is, ones that do not keep track of things by a court reporter or recorder. Appeals from municipal courts of record are not de novo, the higher courts to them, sit as courts of errors like courts of appeals. Motions for new trial and bonds are required also.
In higher courts, an unhappy trial litigant files a notice of appeal in the trial court before the deadline, a timely appeal generally invokes the jurisdiction of a court of appeals. In civil litigation, it is not uncommon for some parties to want to appeal and others not to. Sometimes some trial litigants can hold on to trial rulings they like by not appealing, though other times any single appeal can invoke jurisdiction as against the other parties. Notices of appeal, petitions for discretionary review, and applications for writ of certiorari must generally be timely, otherwise they do not impose appellate jurisdiction. The main exception to this being that the Court of Criminal Appeals may authorize an out-of-time appeal by a writ of habeas corpus (See the post before this one.)
Extraordinary writs, which I have talked about earlier, may give a basis for a higher trial court or appellate court to invoke the jurisdiction of such a court without having started in a lower court.