Once per target successfully hit, regardless of the source of the attacks.
In order to use the Duskblade's 13th level perk, you must take the Full Attack action. It doesn't kick in otherwise. Once you've used it, you get a free spellcast with your full attack action, and then that spell affects each target you hit, for the rest of the round (ie, until you start your next turn). So, let's look at an example.
You choose to make a Full Attack. As a 13th-level duskblade, you have three attacks.
You also invested your feats so as to get Snap Kick, and get another attack off of that. You're using a polearm, so no two weapon feats.
For your three attacks, you stab one guy twice, and trip another guy once (yay polearms!). For your snap kick, you kick someone else.
For the first guy, you land both hits on him. He gets affected by the spell once (one for each target you hit, even though you hit him twice). You miss the snap kick, so no spell damage there. You hit on the trip, so you get spell damage there. (Trip involves making a touch attack. The touch attack was successful, so the spell goes off.) You also have Improved Trip, so you get to hit him while he's down, but that won't give you any extra spell damage because you already connected once with the trip attempt.
Again, you only get one instance of spell damage per target for the turn.
Your turn ends. Someone else's turn starts. They provoke an opportunity attack from you. It's still the same round, so your spell is still charged. If you hit them, and they haven't been hit by the spell damage yet this round, you get to hit them with it too.
Your next turn starts, the spell discharges, and if you want to keep the fun rolling, you'll need to use another full attack and spend another spell slot.
So, actually, it combos reasonably well with tripping... as long as you like the idea of tripping a bunch of different people each round.
Alternate Rules Interpretation: once per successful attack, regardless of source.
There are a number of people who might interpret the rule to say that you deal spell damage every time you hit someone. Even if that's not the rule, going to 13th level as a Duskblade is enough of an investment, and far enough from the paths of True Power that your DM may decide to houserule it in your favor. (As a DM, I probably would.)
In that case, it becomes significantly better, because you get spell damage off of every hit. That means that if you trip someone, and then hit them when they fall, yes, you get to blast them with spell damage twice. It's two independent hits (one touch attack, one normal attack) and that means the spell damage kicks in twice. Even better, you get even more abuse out of that old tripper mainstay of using opportunity attacks (perhaps a trip attack) to punish your just-tripped foe for standing up.