The Basic Rule
The basic idea behind Slow Rests is that a short rest is one night's sleep (8 hours), and a long rest is one tenweek (10 days) of straight down-time.
The immediate ramifications of this have mostly to do with spells and with healing.
Spells suddenly become more significant effects. It takes one week of study and preparation to set up another spell to be cast, and each spell represents several days of work.
Healing becomes rarer and more significant as well. Most HP loss can be described as some mild injury, as it will take time to restore that.
These are cosmetic effects, but they will likely have an effect on how the players approach a problem, avoiding using limited resources, like spells and HP, to overcome a challenge if possible.
Module: Gradual Healing
If a long rest is a tenweek, it can make sense for that week to be interrupted at various points, out of necessity or disaster. A character doesn't just gain all their HP back at once on day 10, they gain it back gradually over the course of the week. The same is true with spells.
Largely, this can mean that each day during a long rest, a character heals 1/10 of their HP, and recovers 1/10 of their spell slots (starting with the lowest-level slots). A 1st-level character with 10 HP and 2 1st-level spells, for instance, would recover 1 hp/day, and recover 1 spell every 5 days.
The character might also spend their Hit Die at any point in the week to gain that HP boost, if they so desire.
Module: Random Encounters
If your players decide to rest in somewhere that's not exactly safe, there's a good chance that something will come along and interrupt their rest in some way. This something is a Random Encounter.
A Random Encounter here isn't always necessarily hostile, but it may turn hostile, and if it does, the healing is cancelled. If it doesn't, an extended rest can go on normally. For encounters that are not hostile at all, you may want to see the Events module instead.
A location's Encounter Rate is the number on 1d20 that, if you roll more than it, you have an encounter. To determine if a random encounter happens during a long rest, roll 1d20, and if that roll is under the Encounter Rate, there is an encounter at some point during the rest.