As far as I understood from the Rules of D&D 5e attack spells require an attack roll against the enemy's armour class, like a normal melee or ranged attack.
Question: Against which difficulty class are heal (or other no damage spells) rolled? Or are those automatically successful and only the healing amount is rolled?
Spells do exactly what they say. Nothing more, nothing less.
There are a few different kind of spells, but I'm going to explain 3 basic ones and that should cover the bulk of them:
The first and most prevalent kind are the healing spells you mention. These, and most other buffing spells actually do not require an attack roll or saving throw, they simply happen. If they indicate a dice expression, you roll that and apply it the way that it says to. For instance the Healing Word spell has you roll some d4s (the number depends on the casting slot), and add your spellcasting ability modifier. You don't make an attack roll, the target doesn't make a saving throw, you just apply the healing to the target.
The second most common type of spell is a debuff or damage spell that has a saving throw attached. For this kind of spell you only roll the damage dice (if there are any), and the target(s) of the spell roll(s) a saving throw of the specified type against your spell save DC for the spell (though generally in reverse order, have the targets roll their saves first, then you roll the damage).
The third kind of spell involves an attack roll. For This you roll a d20, add your spellcasting ability modifier and proficiency bonus for that spell and then compare it to the target's AC. Then you roll any damage or apply any effects specified in the spell.
These are the 3 basic types of spells, and pretty much every spell falls into one of these three buckets. Almost all healing spells fall into the first bucket, so you'd simply roll the dice it says to roll and apply it to the target(s) of the spell. It's also important to be aware of any additional modifiers to those numbers (like the Preserve Life class feature of a Life cleric).
The spell description describes the effect and mechanics of the spell.
For cure wounds, for example, it states:
A creature you touch regains a number of hit points equal to [..]. This spell has no effect on undead or constructs.
It does not mention any saving throw or attack roll. Therefore, by rules as written, it is automatically successful when cast on a creature that is neither undead nor a construct.
The DM however is free to house-rule that unwilling creatures get a save.
To clarify wax eagle♦'s post, the three types of spells in 5e would more accurately classified as follows:
Attack Roll Spell: This spell requires the caster to make an attack roll versus the target's Armor Class (AC). If the attack roll succeeds, then the spell affects the target.
Saving Throw Spell: This spells requires the target to make a saving throw versus the caster's Spell Save DC. If the saving throw succeeds, then the spell does not affect the target. (Note that some spells may partially affect the target on a successful saving throw)
Unopposed Spell: This spell automatically affects the target.
The Basic Rules can be found here.
As mentioned by jkat718's answer and wax eagle's answer, the mechanics that determine whether spells actually take effect fall into three categories:
The first two categories are determined dynamically and apply exclusively to spells that would normally be considered offensive, but last category (under which healing spells fall) can apply to both friendly and offensive spells: consider Power Word Kill (Basic Rules, p. 98) which requires a target with less than 100 HP, but not an Attack Roll or Saving Throw. For all spells in this category (including healing/friendly spells), the spell manifests if its requirements are met, with no RAW chance to resist.
However, a character could reasonably try to avoid such a spell's outcome by avoiding the spell entirely. An obvious way would be by blocking the spell with Counterspell (Basic Rules, p. 86) but this method gives us a clue as to other ways the spell might be prevented; Counterspell's casting time is:
1 reaction, which you take when you see a creature within 60 feet of you casting a spell
You have to be aware that your target is casting a spell in order to counter it. It therefore stands to reason that if you were aware someone was casting a spell you might do something else to interrupt them, such as move out of range, providing you have a reaction (which I'd usually rule you have out of combat) and taking such an action would not take longer than the casting time of the spell in question.
For example, Cure Wounds (Basic Rules, p. 86) has a range of touch, a casting time of one action, and uses verbal and somatic components. It would therefore be obvious enough that some sort of spell is being cast. If you had reason to try to avoid it, you might try to avoid being touched - probably prompting a dexterity skill contest.
Perhaps a more likely example would be Geas (PHB, p. 244) where a DM could reasonably rule that if the target notices the verbal components of the spell, they could attempt to get further than it's 60 foot range away before the casting time of 1 minute has passed (or perhaps cast Silence on the caster before they can complete those components). This is, however, more contentious, as there's no stipulation that the target must remain within 60 feet for the duration, but if the target of Cure Wounds notices in time to avoid being touched at all, then even RAW the spell cannot be cast.
