First, the name "circuit" comes from the old practice of judges "riding circuit." Their circuit was the geographic territory assigned to them. They'd be responsible for hearing appeals in that limited geographic region, and they'd literally ride around the area (on an established schedule) hearing appeals from cases that came up in different parts of their assigned area. It was a way of providing easier access to appellate courts than requiring everyone come to one centralized place in the region.
Second, about the structure of the courts: in the federal judiciary, there are federal district courts (roughly, where trials happen), federal circuit courts, and the Supreme Court. There federal circuit courts are courts of appeal that sit one level above the district courts. There are 13 federal circuits. There are the First through the Eleventh Circuits, as you note. There's also the U.S. Court of Appeals for the District of Columbia Circuit (the D.C. Circuit), as well as the U.S. Court of Appeals for the Federal Circuit (the Federal Circuit). That last one is extremely unhelpfully named, since they're all "federal circuits." The Federal Circuit hears certain types of cases specially designated for it to hear (for instance, patent appeals and appeals from two lower federal courts established for specialty jurisdiction). The numbered circuits and the D.C. Circuit are geography-based. If you're wondering why D.C. needs its own circuit, consider the metric fuckton of administrative rules and regulations made by the federal government. That mainly happens in D.C. It's more complicated than that, of course, because Congress will write into certain statutes that the D.C. Circuit has jurisdiction over appeals of challenges to those statutes. Other times, litigants may choose to sue an agency in one of the other regions.
Third, the 5th Circuit is unquestionably and unarguably conservatively aligned, just as the 9th Circuit used to be unquestionably and unarguably liberally/progressively aligned. The label is decided by looking at both outcomes and reasons given for those outcomes. Pick a hot political topic. Pick a legal case involving that topic in some way. It's extremely predictable how the 5th Circuit will rule. None of this is controversial. The controversy would come from taking the next step: if you say, "Well, they're mostly right." That makes you a conservative, and that enlists you in political controversy. Because as a factual matter, they have a historically fairly high reversal rate (i.e., among the 5th Circuit cases the Supreme Court grants cert. on, the 5th Circuit gets reversed ~80+% of the time), so they're not right in the sense that the Supreme Court agrees and thus their rulings are usually the law of the land.)