Why Czech Republic has always the suffix "Republic"?

There was a country called Czechoslovakia. which split into two parts. the county you ask about, and Slovakia. Presumably, it was thought that the name 'Czech' alone sounded a little odd, so the suffix 'Republic' was added. Remember the name 'Czechoslovakia' was, itself, only invented in 1918 after the area obtained freedom from the Austro-Hungarian Empire at the end of WW1. The Czech people live in the area which earier was called Bohemia and Moravia and the Slovaks in Slovakia.