Apostille is endorsement of an official document (such as a birth certificate) for international use. It's usually a stamp, although it may be a separate piece of paper. Only certain countries issue apostilles.

Apostilles are issued by the office/authority, which issued the original document. In the United States, apostilles are usually issued by the local Secretary of State, which issued the document that is to be certified.
Many countries, incl. Slovakia, do not recognize official documents issued by foreign countries, unless they bear apostilles.
Most Western, incl. all EU member countries issue and accept apostilles, with the notable exception of Canada. You may find an up-to-date list of countries here.
No, since 2019, all EU countries, incl. Slovakia, recognize each other's documents without the need for apostilles.