When I, for example, cast searing smite as a paladin, and then attack and hit successfully, does the spell end after this successful attack, or does the smite last until the concentration ends or the user ends it?
They only do extra damage on a single hit, but most of them have an ongoing effect for the spell's duration
Paladins have access to a total of 7 smite spells (all in the PHB):
- Banishing smite (p. 216)
- Blinding smite (p. 219)
- Branding smite (p. 219) - the only one freely available in the SRD
- Searing smite (p. 274)
- Staggering smite (p. 278)
- Thunderous smite (p. 282)
- Wrathful smite (p. 289)
Let's use searing smite as an example, as you've mentioned. Searing smite is a concentration spell with a duration of up to 1 minute. The spell description says:
The next time you hit a creature with a melee weapon attack during the spell’s duration, your weapon flares with white-hot intensity, and the attack deals an extra 1d6 fire damage to the target and causes the target to ignite in flames. At the start of each of its turns until the spell ends, the target must make a Constitution saving throw. On a failed save, it takes 1d6 fire damage. On a successful save, the spell ends. If the target or a creature within 5 feet of it uses an action to put out the flames, or if some other effect douses the flames (such as the target being submerged in water), the spell ends.
The effect only ever applies to one creature - the first one you hit with a melee weapon attack while you're concentrating on the spell. However, searing smite has an ongoing effect that continues affecting that creature for the duration of the spell (though it specifies that if the creature passes the Con save or someone douses the flames, the spell ends as well).
All of the smite spells do some extra damage on a hit (specifically, the first hit with a melee weapon attack - or with any weapon attack, for banishing smite and branding smite - while concentrating on the spell). As noted in Eddymage's answer, an official ruling in the Sage Advice Compendium reiterates that this extra damage only occurs once, the next time you hit with a qualifying attack after you cast the spell.
Most of these "smite" spells - banishing smite, blinding smite, branding smite, searing smite, and wrathful smite - also have an ongoing effect that lasts the duration of the spell.
The other two "smite" spells have no ongoing effect. Staggering smite forces a Wisdom save; on a failed save, it gives the target disadvantage on attack rolls and ability checks, and the inability to take reactions, until the end of its next turn. Thunderous smite forces a Strength save to avoid being pushed 10 feet and knocked prone. The effect of the former has a set duration, while the effect of the latter just occurs instantaneously.
The extra damage on hit applies only once. The other effects depend on the spell.
Does the blinding smite spell deal extra radiant damage on every successful attack I make while I’m concentrating on the spell, or only on the first one? “The next time” in the spell description indicates that the extra damage applies only once, the first time you hit a creature after you finish casting the spell.