Before the split with Slovakia in 1993, the country was called Czechoslovakia (Czech: Ceskoslovensko). The official name changed with time - Czechoslovak Republic, Czechoslovak Socialist Republic or Czechoslovak Federative Republic. Foreigners often mistook Czechoslovakia with Yugoslavia, could not spell it right, could not even pronounce it, made jokes about it (there is a nice episode in In Last Of The Summer Wine: Compo remembers a guy back in school who whenever needed a reason to beat the other kids, so he just asked them to say Czechoslovakia and beat them if they could not pronounce it correctly), but generally everybody was used to it.

The problem came after the split. The official name was easy - the Czech Republic (Ceská Republika). But what about a short name for everyday use? There were several candidates -- Czech Lands or Czechlands, Bohemia, Czechia; in Czech: Ceské zeme, Cechy, Cesko and few others were discussed.

The Czech name seems to finally settled down to Cesko. Not everybody was/is happy (some people still say it does not sound nice) but the name is used more and more often. The German equivalent Tschechien (Tschechei has bad connotations from WWII) is commonly used. I do not know about frequency of Spanish Chequia or French la Tchèquie, but English Czechia is rarely used. Instead, the full name or simply Czech is used. Strictly speaking, Czech is supposed to be the name of the inhabitants of Czechia or the adjective related to it, but somehow, native speakers of (at least American) English do not seem to care and use it as the name for the country as well.

Why do we need a short name? First, with few exceptions like Dominican Republic, most countries have one. One usually says France, and not the French Republic, similarly Mexico and not United Mexican States. Second, the Czech Republic exists only since 1993, and the Czechoslovak Republic since 1918. However, the Czech nation, society or whatever you want to call it is around for more than a thousand years. It is kind of funny to say that the most admired historical figure of the Czech Republic is Kind Charles IV.

Note: the first C in all the Czech names of the country have a wedge (Česko in case you browser supports the encoding) See also czechia.org