Arial Black is completely right. Divine Smite can be declared at any point prior to rolling the damage dice, and ALL dice rolled for a given attack are doubled (edit: IF it's a critical hit).
This means that, if a Paladin is 11th level or higher, you can get something like this:
Wielding a 1d8 weapon (say, a rapier, with a 16 Dex). Normal weapon damage: 1d8+3. Improved Divine Smite adds an additional 1d8, so the damage for a single attack is now 2d8+3.
Paladin is hitting an undead creature (say, a vampire), and lands a critical hit. The Paladin chooses to make this hit a Divine Smite, and spends the highest-level slot he possesses (at 11th level, this would be a 3rd level slot). A 3rd level slot would give 2d8 (base) + 2d8 (2 levels higher than 1st) = 4d8 bonus damage; however, because the enemy is undead, there is an extra +1d8 damage as well, for a total of +5d8.
Thus, if the attack were a normal hit, it would deal 7d8+3 damage. However, because it is a critical hit, you roll every damage die twice, and use both. So the Paladin will do 14d8+3 damage; rolling that many dice means the value will hew very close to the average, which would be 14*(4.5)+3 = 66 damage, of which 2d8 was piercing damage and 12d8 was radiant damage. (For convenience's sake, I would declare that the first two dice rolled were from the weapon itself, and all other dice were radiant damage.)
This would be further increased if the Paladin had a magic weapon that dealt extra damage per hit. For example, a Flame Tongue sword deals +2d6 fire damage (I can't recall if it has a + on it or not), so the attack would do 2d8 piercing + 12d8 radiant + 4d6 fire + 3 (might be +4 or 5, if the Flame Tongue sword has a +), for an average of 9+3(or more) piercing + 54 radiant + 14 fire = 80(ish) average damage.
Of course, the Paladin in question can only do this 3 times a day (having only 3 third-level slots), and that's if she never spends a single third-level slot on anything but smiting, only smites on a critical, always fights undead, and gets at least 3 crits in one day. Thus, the numbers I'm presenting above are an absolute best-case scenario--the Paladin is highly unlikely to outdo that. If the target isn't undead (or a demon), you'd lose 9 (2d8) damage right there, for an average damage of 57 with a plain, nonmagical rapier.
(Some, but not all, undead are vulnerable to radiant damage--which would push these numbers even higher. I don't know if there are any beings that have resistance to radiant damage, but if there are, they would be this Paladin's bane, as the majority of the damage she deals is radiant.)