Epona is Adult Link's horse.  You can use her to ride around Hyrule field much quicker.  She can also jump over the broken bridge at Gerudo Valley, and is your only means of finding Big Poe's. 

To get Epona follow these steps:

1) Learn Epona's Song from Malon at Lon Lon Ranch when you show her your ocarina. Can only be done some time after meeting Princess Zelda and only as Young Link.

2) Become adult Link and have at least 70 rupees.

3) Talk to Mr. Ingo outside the corral at Lon Lon Ranch and pay him 10 rupees to ride a horse. Instead of getting on the horse provided, play Epona's Song and Epona will run up to you.  Ride her around the corral until the time is up.

4) Talk to Mr. Ingo again and pay to ride again.  Once again play Epona's Song and ride Epona.  But this time, walk up to Mr. Ingo while on Epona and use Z Targeting to talk to him.  He'll challenge you to a race with a 50 rupee wager.

5) If you win the race, Mr. Ingo will challenge you to a second race.  Win again and you will get Epona.

6) To get out of the ranch, get Epona running really fast (by using the A button) and jump over any fence.

Racing Tips: 

Although racing Mr. Ingo can be tough, here is an easy way to win.  Hold forward on the control stick as soon as the race begins.  Use 1 carrot (push A to use a carrot) and wait for it to come back before using it again to keep up with Mr. Ingo.  Hug the fence and wait for him to move out of your way when he takes the corners wide. Then use up all the carrots you have EXCEPT ONE to speed up and pass him.  Then wait until a second carrot appears and use it, then keep doing this to keep ahead of Mr. Ingo.  NEVER USE YOUR LAST CARROT!  Just keep using the second-to-last carrot.  If you use the last one you'll slow down big time and almost definitely lose.

Tip: 

You can play Epona's song whenever you're on Hyrule Field now to summon her to you.  You can also now wake up Talon in the house in Kakariko Village to get him to return to the ranch. This allows Link the opportunity talk to Malon, while riding Epona, and try an obstacle course so Link can win his very own cow. It will appear in Link's house in Kakariko village.