Chocobos are gigantic birds, mascots of the entire Final Fantasy franchise and are basically used as a transportation method. It's not different in the case of Final Fantasy XV - after a few hours you will be able to unlock access to them, allowing you to travel around the world significantly quicker, especially because you're not able to get everywhere with the Regalia.
To unlock access to Chocobo, you have to complete one side quest. You will gain access to the quest at the beginning of chapter 3. After reaching Duscae Prompto will be excited about the possibility to visit a Chocobo farm and will ask if you can travel there. Accept his request and a new marker will appear on the map, or to be more specific at Wiz Chocobo Post. Head there to begin the quest.
Friends of a Feather
Recommended level: 15
Reward: Unlocking Chocobo
You have to travel to Wiz Chocobo Post and approach one of the characters there, Wiz - marked with a yellow question mark on the map. Once you're there, a conversation will start automatically. From it you will learn that renting Chocobo's is temporarily out of the question, as the nearby area is terrorized by a powerful Behemoth. You will be asked to take out the beast and the quest will begin. End the conversation and speak with Wiz again, select Hunt and start "A Behemoth Undertaking" hunt. You should also eat a meal at Wiz's or at camp before you set out - it will make the quest a little easier for you.
Your task is to find the lair of the beast - head towards the marker on the map. Upon arrival, the characters will notice multiple trees falling down on the ground - it means that you're getting closer to the target. Continue your march to a number of markers on the map. Soon you should stumble upon a group of Veretooth - take them out and continue your journey, heading into each branching to collect a large amount of items. You should soon reach a place filled with thick fog that severely reduces visibility.
You will be able to see a huge beast in the fog - that's Deadeye, the monster you have to take down. Your task (for now) isn't to engage it in combat - you have to follow the beast and prevent it from noticing you. Use the stones scattered around the area to hide from the beast's eyes and continue following it. You can't get too close and at the same time allow the beast to move too far away from you - in both cases you will be alerted by the game. Keep following the beast and you should soon reach a narrow passageway between rocks. Get to the other side.
Use a nearby fire deposit to create a fire spell - the enemy you're going to be fighting with is extremely vulnerable to that element. However, spells aren't a necessity to take down Deadeye, so you shouldn't worry too much about it. Continue towards the marker and you should soon reach the beast, and the battle will commence.
The game will inform you that you can use nearby barrels. Detonating them will deal huge damage to the beast, but Deadeye will be jumping around the whole area, making it next to impossible to the beast with it. Instead, concentrate your attacks on the legs of the beast - if you deal sufficient damage, the beast will fall on the ground and enter "vulnerable" state, allowing you to deal huge damage to it (and maybe detonate a barrel) before it gets up again. Remember that Blade of Brennaere deals fire damage - with a 20 level character you will be able to deal up to 5,000 points of damage with a single attack.
Continue your attacks and the beast should soon be lying dead on the ground. You can now go back to Wiz. Before you do it, however, head to the elevation from where Deadeye entered the arena - there's a Broken Harmonica there. Afterwards, head to Wiz, start a conversation and the quest will end.
You now have to approach one of the Chocobo Rentals (a bird icon on the map) to rent a Chocobo. The price is extremely low (especially for the benefits), which is why it's always worth it to purchase the maximum amount of days - 7 - for which you will pay only 350 Gil. Now you have to press and hold R2/RT (PS4/Xone), press left/right on the d-pad and select "Call Chocobo" option. You can now mount the bird and use it to travel around the world significantly faster than on foot. The amount of remaining rental days can be checked on the upper right corner of the screen, to the left of the mini-map.
A detailed description of Chocobo development, available food types and customization options can be found in a separate chapter.