I recall some rules in earlier editions where if you miss with a touch spell, it's "held" and you can try again on a future turn. Does that rule exist in 5e, and if so, where it is?
I'm guessing I'm just missing it, but I couldn't find it in the PHB.
In general, if you need to touch the target, this is handled as part of casting the spell, with a melee spell attack to determine if you succeed. For example, shocking grasp:
Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target. You have advantage on the attack roll if the target is wearing armor made of metal. On a hit, the target takes 1d8 lightning damage, and it can't take reactions until the start of its next turn.
If the spell attack fails, the spell does nothing. Note, since you're used to the mechanics from 3rd edition, that spell attacks are rolled against the target's normal AC.
There are a few exceptional cases:
This includes flame blade and vampiric touch. These also use melee spell attacks, but the spell lasts for some duration and allows you to make additional attacks during that time.
The various paladin "smite" spells such as branding smite are all verbal-only bonus action spells (so you can cast them with a weapon in hand and immediately attack) with a duration of 1 minute. Once the spell is cast, the next time you hit someone with a melee weapon, the spell adds damage and other effects. This is arguably the most similar to the "held touch spell" from 3rd edition: you make your attack, and if you miss, it carries over to your next attack. However, they're not "touch" spells; you're still making normal weapon attacks.
There aren't really any "touch spells" in 5e, or at least there is no AC modification for the distinction.
What spell are you thinking of specifically?
In general It depends on the spell. If it's a concentration spell like vampiric touch, you can keep trying as long as you maintain concentration until the spell ends. If it's a straight attack like booming blade, if you miss, that's it, you have to cast it again.
P203 of the players handbook explains the above in detail:
Duration
A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours, or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.
Instantaneous
Many spells are instantaneous. The spell harms, heals, creates, or alters a creature or an object in a way that can't be dispelled, because its magic exists only for an instant.
Concentration
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
If a spell must be maintained with concentration, that fact appears in its Duration entry, and teh spell specifies how long you can concentrate on it. You can end concentration at any time (no action required).
