E.g. one has a DT of 10. If I cause 5 damage, he won't get hurt at all, I suppose. What if I cause a damage of 15, 5 more than the DT? Will the DT still be subtracted from the damage total, so will he receive only the amount that's past the threshold, 5 points, or the complete amount of 15 points?

If an attack is less powerful than target's DT, the target will still take 20% damage from the attack. So 5 against 10 will do 0.2*5=1 damage.

If an attack is more powerful than target's DT, the DT is simply subtracted from the attack. So 15 against 10 will do 15-10=5 damage.

Pretty simple, but very effective.

Oh, and if you play on PC with mods with items that have Damage Resistance, the resistance is calculated from the DT result. So with DR50 the target would take 0.5 and 2.5 damage in the above examples.

IIRC, no vanilla item in New Vegas has Damage Resistance, so if you are on console or without mods, ignore that

It also worth to remember that the DT is calculated for each projectile separately. A weapon may have a high DPS but if it does very little damage per shot, it may not be very useful against armored targets.

Use different ammunition smartly. AP rounds usually slightly decrease the nominal damage but the negative DT modifier offsets this greatly when used against armored targets. Hollow points are great against soft targets.

Critical hits only increase the damage or ignore DT too?

Critical hit affect the initial damage, before DR and DT are applied.

And DR is applied before DT:

http://fallout.wikia.com/wiki/Damage_Threshold#Fallout:_New_Vegas

Oh? remembered that wrong then.

And i remember emptying a full Minigun clip into a Remnants Power Armored enemy and him hardly taking any any damage That was before 5mm ammo was patched to have DT piercing.

This. You only do damage equal to what's beyond the DT of a piece of armour. Anything below that threshold is cancelled out.

Remember that even if your DT blocks the entire damage 20% still gets through. Your armour wont take any condition damage, but when you've got a good 8 enemies shooting at you that 20% bleedthrough is still significant.