It should be noted that swords and armour weren't abandoned at the same time, nor were they abandoned quite in the same way or with the same level of entirety, if that makes sense. Its a complex question, so I may simplify somewhat.

In Europe especially but in the world in general, the complete armour sets once worn by wealthy heavy infantry/cavalry was generally rendered obsolete by the innovations in firearms technology, and the increasingly-organised and effective infantry. Traditional knight armour and other traditional suits were ineffective at deflecting musket balls, and cavalry charges lost much of their power with the innovation of organized pike-and-musket formations.

This wasn't the end of armour, however. Thicker and thicker breastplates continued to be worn through the 15 and 1600s (at the expense of full armour kits), until increasingly powerful muskets also rendered them obsolete, too. Cuirassiers, a sort of Napoleonic heavy cavalry, continued to wear breastplates capable of deflecting a musket pistol shot or a sword blow up until the early 20th century but by the era of line tactics (18th-19th century) almost all other sections of the army had abandoned the wearing of armour.

Swords continued to be used until the 20th century, but from the 18th century onward they became more and more a weapon exclusive to infantry officers, artillery crew, and cavalry. Regular infantry used bayonets on the end of their muskets from ~18th century onward, while before then pikemen (who wore but didn't often use swords) provided the "melee" element of the infantry.