AceCalhoon's answer is only mostly correct. While counterspell can be blocked with another counterspell, a single wizard could not counterspell a counterspell that targets a spell he is casting. Although he can take reactions, he cannot actually cast counterspell in the middle of chanting fireball.
Wizard One attempts to cast fireball.
Player Wizard uses counterspell.
Wizard One cannot counterspell back because he is in the middle of casting fireball.
Both fireball and counterspell have verbal components. According to the rules of components on PHB 203, "If you can't provide one or more of a spell's components, you are unable to cast the spell."
You need multiple wizards for this, or a sorcerer with the subtle spell Metamagic.