Basically, "normal" combat works on a round-by-round basis: each round one or the other of the two units will "win" and inflict damage on the other unit. The chance of each unit winning each round, and the amount of damage that they do is determined by their relative strengths.
So, for instance, you might get:
Axeman vs Chariot
Combat Round 1 - Axeman wins, inflicts X damage on Chariot
Combat Round 2 - Chariot wins, inflicts Y damage on Axeman
Combat Round 3 - Axeman wins, inflicts X damage on Chariot
Combat Round 4 - Chariot wins, inflicts Y damage on Axeman
Combat Round 5 - Chariot wins, inflicts Y damage on Axeman
Combat Round 6 - Chariot wins, inflicts Y damage on Axeman
Combat Round 7 - Axeman wins, inflicts X damage on Chariot
Combat Round 8 - Chariot wins, inflicts Y damage on Axeman, kills Axeman
Chariot wins the battle with some (X+X+X) damage.
Now, the first strike is basically a "free shot" round in which one of the units gets an extra chance to inflict damage. In a "first strike round", the percentages of each unit hitting the other are as usual, but there's a difference -
the unit with the first strike is the only one that can inflict damage during that round.
To illustrate by example:
Archer vs Chariot, Archer has 2 first strikes
First Strike Round 1 - Archer wins, inflicts X damage on Chariot
First Strike Round 2 - Chariot wins,
no unit recieves any damage (ie "draw")
Combat Round 1 - ...etc
So the first strike rounds are "freebies" in which the unit with the first strike(s) is itself invulnerable to damage, but may inflict damage to the other unit. After the first strike rounds, battle progresses in the normal way with combat rounds in which either unit can receive damage.
I hope that clears that up for you. Just incidentally, I think that generally the consensus amongst players is that first strikes are usually not quite as powerful as the combat promotions. However, they're still quite useful for tilting the scales slightly in your favour when the units engaged in combat are fairly
close in relative strength to each other.