If the spell’s controller isn’t the same as its owner (due to Commandeer, for example), it won’t be exiled. The spell is countered and put into its owner’s graveyard. Then the spell’s controller’s graveyard, hand, and library are searched.

If the targeted spell can’t be countered (it’s Vexing Shusher, for example), that spell will remain on the stack. Counterbore will continue to resolve. You still get to search for and exile all other cards with that name.

Counterbore has no effect on cards on the battlefield that have the same name as the targeted spell.

The cards you’re searching for must be found if they’re in the graveyard, because that’s a zone everyone can see. Finding those cards in the hand and library is optional, though.

In most cases, Counterbore will exile the targeted spell. Its first sentence counters the spell, which puts it into its owner’s graveyard. Its second sentence then removes it from the graveyard. However, there are some exceptions to this, as listed below.