It says that the spell targeted disappears if you beat its power. Seems simple enough to me.
Below I have quoted the very first paragraph beneath the Counterspell heading from YS253, please note the portion I bolded.
Counterspell
If a wizard is in the presence of a magical effect, he can attempt to nullify it with the power of his will alone. Even though counterspells are an evocation effect, they can be used to disrupt thaumaturgy. While such an attempt may only be temporary, it can buy the wizard precious time.
The way I interpret that, is that a counterspell potentially nullifies the effects of a spell as it happens (Evocation) or can potentially nullify a spell/effect with a duration (Thaumaturgy) temporarily.
I would also like to point out that the wizard attempting an 18 shift counterspell as described above will also have to take a Severe and a Moderate consequence at a minimum, not two Minor consequences and a Moderate, assuming of course they are successful in summoning 18 shifts of power. A bad roll of the dice would immediately cause the counterspell to fail, and the wizard would still end up taking mental stress...
My reasoning is this. A Counterspell is an Evocation effect, it is essentially described as an attack spell with a target of the spell being counterspelled/nullified. In order for the Counterspell to nullify an 18-shift Ward, that means an 18-shift Evocation effect is required. The RAW on YS255 state:
You can channel a number of shifts in power equal to your Conviction with a minimum of impact—just one point of mental stress.
This is then followed up on YS256 with:
Every shift of power you summon past your Conviction costs one additional point of mental stress during casting;
If the wizard has a Conviction of Superb (+5), then an 18-shift Evocation would cause the Wizard to take 14 boxes of Mental stress. Any specialty in Spirit (Power) makes it easier for the wizard to summon the power, but the wizard still suffers from summoning that much power, since it is still past his Conviction skill level.
In short, the wizard would take some significant harm to nullify the Ward, wouldn't completely negate the Ward, and would at best have a Minor consequence, a Minor Mental consequence, and an Extreme consequence remaining, as well as Physical stress having been taking and possibly Mental stress as well. All before even managing to enter the BBEG's lair, where those consequences the character took can/should/would be used against them...
-Cheers
Edit: Additional comments. From the Counterspelling rules on YS253, there is no mention of using particular elements for the Counterspell. In fact, from the example given, Harry didn't get any Power or Control bonuses to attempt to counterspell, nor was any mention made of what element Harry was using. This would mean that a wizard's ability to counterspell is completely reliant on the wizard's Conviction, Discipline and Lore skills, as well as the number of stress boxes and consequences the wizard has available and is willing to use. I mention Lore as well because even if the wizard was to use the Sight to see the Ward, they would still need to make a Lore check to correctly determine how much power was required to counterspell the Ward.