Inspiration is carried by the player and can be transferred from character to character. If one character falls, another may be inspired to rise up. Characters start with no Inspiration whatsoever. Inspiration can only be earned by playing.
Whenever you play out one of your character's personality traits, follow your character's ideal, or give in to one of your characters drawbacks or bonds, you gain +1 Inspiration. If you do this to your character's detriment, you gain +2 Inspiration. If you do this and it gets your character killed, you gain +3 Inspiration. Your character can never have more Inspiration than their current level. Excess Inspiration is automatically lost.
For example, Bob is playing Robin, a 7th-level Fighter with the Noble background. Robin has the Bond "My loyalty to my sovereign is unwavering." Robin overhears mercenaries who work for Robin's sovereign plotting to desert from their posts before a battle. If Bob intervenes or confronts the mercenaries he could gain 1 Inspiration, if it causes him to lose an ally or suffer greatly he could gain 2 Inspiration, and if Robin gets killed confronting them Bob gains 3 Inspiration. If Bob waits to report the mercenaries to a commanding officer then he gains no Inspiration, he didn't handle them himself.
One point of Inspiration can give you advantage or take away disadvantage on a single d20 roll.
Three points of Inspiration can turn one of your unsuccessful d20 rolls into a successful rolls.
When your character dies you roll up a new character at 1st-level and lose all of your Inspiration. Each point of Inspiration you have raises the starting level of your new character by +1. If you were holding onto a maximum amount of Inspiration then your new character would start at the same level as your old character.
For example, Charlie has 5 Inspiration and is playing Tytus, an 8th-level Cleric. When Tytus dies, Charlie rolls up a Rogue, Marigold, and she starts at 6th-level.