Strike simply means to deal damage equal to the unit's power. For example, if a Bosk Troll strikes, it strikes for 3 damage. In a fight, units strike each other simultaneously. Some abilities allow units to strike first, e.g. Lance. Split power units deal damage equal to their attack power when they strike. Sometimes the striking unit and the unit being struck are at different locations, and some effects (e.g. Mask of Mayhem) care about the location of the strike. For clarity, the strike occurs at the location of the unit being struck.
At the moment a strike is resolving against multiple units, the controller of the striking unit declares how to split up the damage among the struck units. The damage may be split up however you prefer, including assigning zero damage to some units (see below). The total damage assigned must equal the total power of the strike. Units assigned zero damage are still considered struck, and the striking unit makes a total of one strike, split among multiple units. The damage is inflicted simultaneously on all units.
A strike for zero damage, e.g. Grandmaster Wizard, is still a strike, even though it doesn't do damage. See the 'zero' codex entry for more details.
Some effects care about a unit's 'next strike' or 'the next time it strikes.' If the unit is making multiple strikes at the same time (from an effect like Whirling Blades or Leap Attack) then all of those strikes are considered the 'next strike.' This means that a unit with a Lance that is affected by Spin Attack will deal one extra damage on each of it's strikes. Note that this is different from attacking while there are multiple defenders. In that case, the attacker makes a single strike, and the damage is split among the defenders.