In N. America and Australia the most common hive is the Langstroth hive (rectangular w/cold-way frames). In Brittain (and Ireland?) the most common is the National (square, either cold or warm way) while the rest of Europe has pretty well standardized on the Dadant/Blatt hive (rectangular, cold-way).
Other common hives are Top-bar hives. These are used commercially only in Africa, AFAIK, but plenty of hobbyists keep these in N. America and Europe. These are horizontal hives (you don't stack boxes) without frames. Cold/warm way depends on where you cut the entrance.
Picking up a little steam in Europe and N. America is the People's Hive ("Warre"), which is a square hive with top-bars rather than frames.
Management techniques of framed-hives (Langstroth, National, Dadant) are all virtually identical. Management of top-bar hives and the Warre is wildly different from framed hives, though any framed hive could be managed as if it were a Warre.
If you are starting out and are budget-strapped, I recommend building a top-bar hive or Warre. Otherwise, you are best off using one of the standard framed hive types because it is most familiar to other beekeepers in your area from whom I expect you will be getting some advice and/or assistance.