Most spells are cast instantly and their effect is worked out at once, for instance a 'Fireball' is cast, it strikes a target, and damage is resolved. In this case the spell card is returned to the player's hand, and, although it cannot be cast again that magic phase, it can be cast afterwards as normal.
Some spells last for longer than this and these are said to either Remain in Play or Last One Turn.
Remain in Play
Once cast some spells continue to have an effect from turn to turn. These spells are said to Remain in Play and this is indicated on the spell card itself. When the spell is cast the card is placed beside the target to indicate that it is affected by the enchantment.
An example of a spell which remains in play is the Battle Magic spell Net. When this spell is cast the target is fixed in place by a magical net and cannot move until the Net is dispelled. The card is placed next to the target to indicate that it is netted.
Last One Turn
Some spells have an effect which lasts for the duration of a turn. These spells last from the moment they are cast until the start of the casting player's following magic phase. This means they will last throughout the enemy's turn and for the movement, shooting and close combat phases of the caster's following turn also.
Spells which last one turn are automatically dispelled at the start of the player's following magic phase and can be cast again that phase. It is good practice to remove all such spells at the very start of the magic phase, as this will save confusion later.