County Court as a Appellate Court

The Texas Constitution provides that every county in Texas shall have a county court. In rural counties, the county court is a real court presided over by the elected county judge. Such a county judge need not be a lawyer, and is not only the chief judicial officer in the county, but is also the chief executive of the county (though with less hands-on power than any of the county commissioners, since the judge does not have a precinct to have power over and responsibility for) and the chief legislator of the county presiding over commissioners' court, which is the county legislature. (Technically district court judges, who, as judges, outrank county judges are not judges of the county, but are independent of the county. In rural areas, it is common for a district court to encompass more than one county, and the district judge to ride a circuit among a number of counties). One of the jobs of the county judge's judicial county court is that it is the appellate court for justice courts and small claims courts in the county. These appeals are trials de novo.
Larger Texas counties have one or more county court at law which are established by the legislature. Such counties don't have judicial county courts, the county court at law performs all the judicial functions of the judicial county court. In such counties, the county judge is not really a judge; the title is a courtesy since such a person's most important role is to preside over commissioners' court. County courts at law generally must be presided over by a lawyer. Such county-courts-at-law take the appeals of the justice and small claims courts of the county, generally by trials de novo. Municipal courts that are designated by the legislature as municipal courts of record have appeals by reporter's record and clerk's record like a district court, and the county court at law sits and rules as an appeals court justice does.
Appeals by writ of certiorari is sometimes possible from a justice or small claims court to a judicial county court or a county court at law.
Except for constitutional violations, there is no appeal from a judicial county court or a county court at law sitting as an appellate court over a lower court.