Jwitti's reply is a good approximation. If you want more detail (and you may well not want it), read on...
All combat in Civ IV is conducted in rounds. The attacker and defender each have a chance to win each round, and if they do, they'll deal a little damage to their opponent. The chance to win and the damage are based on the units' relative strength. This continues until one unit is dead (usually).
If one unit has first strikes, or has more first strikes than its opponent, then for the first x rounds, where x is the number of first strikes, that unit won't take damage. For instance, an archer with one first strike is defending against an axeman with no first strikes. If the archer wins the first round, he damages the axeman as you would expect. But if he loses the first round, he'll take no damage. After the first round, the loser always takes damage.
If the archer had Drill I (+1 first strike, for a total of two), then he wouldn't risk any damage for the first two rounds, but then he'd be vulnerable after that.
For full, excruciating details, see
Combat Explained in the War Academy.