Sheriffs in the English tradition are utterly unlike American sheriffs. American sheriffs are essentially enforcers of the court: serving process, running the county jail, courtroom security, etc. In rural areas they are the primary law enforcement. In urban areas they typically relegate law enforcement to a municipal police department, but I'm made to understand that Maricopa County in Arizona is geographically large and that the sheriff in question has substantial law-enforcement obligations (that he is presently accused of shirking).
In American popular culture, sheriffs are often dismissed as country bumpkins who are invariably lazy or corrupt. In sparsely inhabited counties (such as those where I live), the sheriff is often simply the only citizen who decides he wants the job; he may have little or no actual law-enforcement expertise. In Sheriff Joe's case, the auto-appellation "Toughest Sheriff in America" is somewhat backhanded, because many of this sheriff's charges are jail inmates with essentially no rights or power to resist, and he stands accused of brutalizing them while under his care. It's like a sixth-grader calling himself the toughest kid in school when all he beats up on are the special-needs students.
As I explained several pages ago, the reason America preserves the sheriff office is that a state in the United States is too big a region to administer effectively for things like court matters and local law enforcement, and not everyone lives in city limits where a municipal government would apply. The county is a well-sized administrative unit, and hence every resident of the United States falls under the jurisdiction of a local sheriff who reports to a local district judge or county executive.