To counter (打ち消す uchikesu or 打ち消された uchikesareta) the effects of a spell or PIECE card is to make it so that those effects do nothing. The cost of the card is not refunded. The spell or PIECE card is still considered to be used, so effects that trigger whenever a spell is used such as that of Code Piruluk APEX will still trigger.
By necessity, cards and abilities that counter the effects of spells have Use Timing [Spell Cut-In]. Triggering the Life Burst of a spell does not count as using the spell, so a player may not use a card or ability with Use Timing [Spell Cut-In] in response to those. As card with Use Timing [Spell Cut-In] are used after the spell's targets and modes are chosen, countering is done after the spell's targets and modes are chosen.
The vast majority of countering effects counter the effects of spells; for a list of such cards, see Category:Counter Spell. Only one card, Eternal♡Immortal Kyurukyurun☆, can counter the effects of PIECEs.