Bardic Inspiration, not to be confused with DM Inspiration, is an ability all bards have in 5th edition.
On your turn you can use a bonus action to use Bardic Inspiration on another creature (you cannot use it on yourself). That creature is then Bardic Inspired for the next 10 in-game minutes, or until they use it. A creature can only hold one charge of Bardic Inspiration at a time, but once they use it the bard can re-inspire them.
After making an ability check, attack roll, or saving throw, but before the DM tells the character what the results are, a creature with a charge of Bardic Inspiration may spend that charge as a free action to roll an d6 and add it to the triggering roll.
Almost all the Bardic Colleges expand what Bardic Inspiration can be used on.
What you actually do can be flavored as you see fit, most people that I have played with make it an encouraging word or two.