5th Edition Dungeons & Dragons, or "5e", is a more streamlined version than many previous editions of the popular game. Some changes that have been made are intuitive, and many DMs and players have house ruled such effects for years. Other changes to the game are more radical, being both broader and specific in nature — and the way magic is handled certainly falls into this category. 

Magic spells and their selection used to be a fairly straightforward process. A player would turn to the relevant mage or priest section of the Player's Handbook for their spells and there they would be, listed in level order for a player to choose from. In 5th Edition, however, all of the spells are lumped together and the different sources of magic have effectively been set aside and there are no divine spells or arcane spells as such. This big pool of all the spells is an example of the broader nature of the newest edition of the game. 

Spell availability in 5th Edition is mainly dependent on a character's class. Bards, Clerics, Druids, Paladins, Rangers, Sorcerers, Warlocks, and Wizards all have their own spell lists to choose from, with many spells appearing on multiple lists. Dispel Magic, for instance, cast by a Wizard is identical to that cast by a Druid or Cleric, and their method of acquiring it may vary but it is in effect exactly the same for each. 

Some archetypes must choose from another's spell list with limitations. Arcane Tricksters (spellcasting Rogues) and Eldritch Knights (spellcasting Fighters), for example, choose their spells from the Wizard spell list but are further limited for most of their choices by school (e.g., Abjuration, Evocation). Some archetypes have bonus spells too — always available in addition to their basic spell list provided they have the slots to cast them — and Clerics and Circle of the Land Druids fall into this category. Finally, some characters have extra spells available but are still limited to a maximum number of "known" spells. Warlocks get extra spells to choose from based on their patrons but these must be counted toward their known limit and, similarly, Bards with the Magical Secrets ability may choose from any spell list at certain levels but are also limited by a maximum number of known spells. 

Class spells are either "prepared" or "known," again depending on character class. Prepared spells work similarly to other editions, being chosen from a list of available spells, are changeable after a Long Rest, and are specific to the Wizard, Cleric, Druid, and Paladin classes. Prepared spells to choose from are based on the class archetype: A Cleric receives bonus spells that are always prepared but these vary depending on sphere or "domain"; e.g., a Cleric with the Life domain always has Bless or Cure Wounds prepared but a Cleric of the Nature Domain is more like a Druid, with Animal Friendship and Speak with Animals always prepared, although both choose their normal prepared spells from the standard Cleric Spells list. All other spellcasters have a more limited number of known spells that are locked down, additional spells known are generally granted at each level, and one previously known spell may also be changed at this point. 

In 5th Edition, you do not need to tie to specific spell slots the spells you have revised or know. So it would no longer be, say, "I have three Cure Light Wounds, two Bless, and one Command spell revised," just that you have say six 1st-level spell slots available, four 2nd, etc. You can decide on the fly which spell available you want to cast, plus the slot you cast the spell with, provided it meets the minimum spell slot. Indeed, many spells have more powerful effects when cast with higher-level slots: Fireball, for example, inflicts 8d6 hit points of Fire damage before saving throws, but does an additional d6 for each slot level above 3rd you cast it with (e.g., a 6th-level spell slot Fireball does 11d6 points of damage). 

Classes also define whether or not a character can cast spells with the Ritual tag as a ritual — taking an additional 10 minutes on top of the casting time but not expending a slot. Certain classes may also benefit from a Spellcasting Focus — an item or holy symbol that allows spells without a listed (gold piece) cost to be cast without a material component. Additionally, the "Spellcasting Ability" the spells are affected by is dependent on class — Clerics, Druids, and Rangers use Wisdom; Bards, Sorcerers, Paladins, and Warlocks use Charisma; and Wizards, Eldritch Knights, and Arcane Tricksters use Intelligence. Some characters have racial abilities that permit limited spell use, such as a High Elf's Wizard (Intelligence) Cantrip, and elemental Genasi have racial spells with a Constitution Spellcasting Ability! 

So what does all of this have to do with scroll use in 5th Edition? 

Rules for scrolls are very different from those in many earlier editions and, to understand why, it is important to understand how the magic system works. Magic in 5E is like a great reservoir of water, with different classes tapping into it in different ways — a Wizard might be the equivalent of the owner of a water mill using the power of the water for their own ends, a Druid might draw the water and give it as sustenance to those who need it, and so on. Same water, different uses — and so too with spells. Therefore, scrolls are no longer broken down by class, and a Druid scroll only useable by Druids, a Clerical scroll just for Clerics, and so on, no longer applies. If a spell on a scroll appears on the spell list you use, you can cast it. 

Scrolls are broken down by level, with rarer scrolls allowing higher-level spells to be recorded upon them and crucially using the scroll's Save DC or Attack Bonus rather than the spellcaster's when it is cast (which also increase with rarity) — and whether the player likes it or not! Also, and perhaps because of this greater versatility, spell scrolls only have a single spell written upon them. 

There are limitations of course. If, for example, the spell is on your spell list but at a higher spell slot you have access to at your level, then you must make a Spellcasting check (to which Bards can always add their Jack of All Trades Bonus). The DC is 10+ the spell level (with '"free" cantrips counting as 0 level spells). Wizards may instead copy spells from a scroll into their spell books with a similar Intelligence (Arcana) check, although are limited to the number of Cantrips they may know. 

The upshot of this blurring of the boundaries means that it is entirely possible that an adventuring party might come across a spell scroll that several members could use, despite them having very different classes. As an example, imagine a party of 5th-level characters who have just defeated an evil Light domain Cleric and discovered the following spell scrolls (all of which this particular foe could have used): Faerie Fire, Healing Word, Lesser Restoration, Dispel Magic, and Fireball. 

· Shovelless the Ranger can only use the Lesser Restoration scroll as that is the only spell to appear on his list. 

· Desdinova the Bard can use all of the scrolls except the Fireball scroll. 

· Malyo the Wizard can only use the Dispel Magic and Fireball scrolls. 

· Paros the Arcane Trickster can also only use the Dispel Magic and Fireball scrolls, as he uses the same Wizard Spell List as Malyo. However, until Paros receives a 3rd level spell slot (at 13th level), he would need to perform a DC 13 Intelligence Spellcasting check to use the scroll(s) successfully, as Intelligence is the Spellcasting Ability of an Arcane Trickster. 

More than one class being able to cast exactly the same spell from a scroll goes a long way to explaining why the Read Magic spell that Wizards used to have as standard has been removed — and I won't tell you how long I spent looking for it! 

Having scrolls that more than one class can use is a great tool from a game-designer's point of view and especially useful for a DM to help the characters out of a dead end without it seeming too much like deus ex machina. 

Hope this has been of some help to you! Spell scrolls are explained further on page 200 of the Dungeon Master's Guide.