Here's what it says in the Basic Rules regarding inspiration:
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INSPIRATION
Inspiration is a rule the Dungeon Master can use to reward you for playing your character in a way that's true to his or her personality traits, ideal, bond, and flaw. By using inspiration, you can draw on your personality trait of compassion for the downtrodden to give you an edge in negotiating with the Beggar Prince. Or inspiration can let you call on your bond to the defense of your home village to push past the effect of a spell that has been laid on you.
GAINING INSPIRATION
Your DM can choose to give you inspiration for a variety of reasons. Typically, DMs award it when you play out your personality traits, give in to the drawbacks presented by a flaw or bond, and otherwise portray your character in a compelling way. Your DM will tell you how you can earn inspiration in the game. You either have inspiration or you don't-- you can't stockpile multiple "inspirations" for later use.
USING INSPIRATION
If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. spending your inspiration gives you Advantage on that roll. Additionally, if you have inspiration, you can reward another player for good roleplaying, clever thinking, or simply doing something exciting in the game. When another player does something that really contributes to the story in a fun and interesting way, you can give up your inspiration to give that character inspiration.
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Basically, if you as the DM decides to use the Inspiration system, each player selects (or creates) one or more Personality Traits, an Ideal, a Bond, and a Flaw to help flesh out their character. They help define who this PC is, what is important to them, what they believe in, how they behave, and what things make them act poorly. Then throughout the course of the game, you as the DM can give the player Inspiration (I personally use poker chips) if they play their PC in a way that matches their traits, ideal, bond or flaw. You can also oftentimes
offer Inspiration to the player if they could act in a way that is emblematic of one of their more problematic or issue-creating traits... in order to create drama and interesting new twists in the story. The player doesn't
have to accept it... but it can be fun if they do and they get a small reward for it.
The Personality And Background chapter in the Player's Handbook talk about Inspiration more in depth, and also give many examples players can select from based on their backgrounds on the various traits, ideals, bonds and flaws. And if you want even more details on the kind of depth you can get from this kind of system... D&D inspiration system is influenced heavily from the roleplaying game FATE. If you go check out the FATE system reference document website:
https://fate-srd.com/fate-core/basics
...it goes all into Aspects and how they help create your character's personality and can drive the game. You might even find some of the ways FATE does their system to be more interesting and compelling that the Inspiration method in the PHB. I believe my "offering" of Inspiration example above is more in line with FATE than the D&D's system (as it talks about Compels of a character's aspects.)
For new players to roleplaying, or even just players who've never really thought about creating personalities for their characters (with wants, needs, and desires)... using Inspiration is a nice little way to reward players for doing so.