Well aside from using an axe to cut a pony in half (sorry writing darkfics affects my brain sometimes), the act of hacking is where someone by some means manages to enter a computer network that normally they wouldn't have access to (legally or otherwise). Below is a definition from Oxford Dictionary:

use a computer to gain unauthorized access to data in a system

However, as video games, sci-fi movies and other materials have shown us, this can be achieved in multiple manners.

1. A Computer
By using a computer the character is able to gain access to the system.

2. A Cell-phone
While this is technically just a smaller computer, the one thing that makes cell-phones interesting is that they are mobile. Can be seen in games like Watch-dogs.

3. Magic/Psionics
By using magic (or perhaps through the use of psionic mental powers) the pony is able to manipulate and control electronic devices. Perhaps even use their powers on a computer to hack.

As for what your character can do by hacking, feel free to be creative.

Why Hack?
Why would someone want to hack. Well, generally the common reason is greed as the individual hacks accounts to acquire money either from a bank account or by blackmailing someone else.

However, depending on the world there can be other reasons the pony is hacking into places they shouldn't be. Perhaps they are in a dystopian future and are hacking into the companies higher up and siphoning the money to fuel a rebellion or feed the poor?

Perhaps they have a hunch the the Equestrian government is hiding a dark secret and the pony decides to use their hacking skills in order to bring this terrible secret to life?

It really comes down to your imagination.

Ponies don't have hands
One thing that will come up very often is some one thinking their smart and pointing out that ponies don't have hands. To which I shall congratulate them for being so smart. Your right, ponies don't have hands instead they have hooves.

However, this doesn't mean they can't use computers. With the presence of unicorn magic, touch screens and perhaps even specially desired mouses/keyboards a pony could be able to access and use their computer.

Perhaps there is a mouse that has been designed to function like similar to our mouse but has been designed around the hoof (instead of a hand consisting of a thumb and 1-5 fingers). Or perhaps there is a special mouse designed that a pony can control it using their mouth (yeah that does sound a little weird).

As for the keyboard they would just need bigger keys or a device attached to their hoof for precise keystrokes.

Hacker Terms:
Here are some terms that might be helpful for people trying to write their own hacker fics.

Hacker -

A hacker utilizes alternative system access methods to sabotage computer systems and networks. Hacking actions are differentiated as illegal and unacceptable (black/grey hat hacking), or legal and acceptable (white hat hacking).

(Quoted from Technopedia)

Cracker -

A cracker is an individual who performs cracking, or the process of breaking into a computer or a network system.

(Quoted from Technopedia)

Black Hat Hacker -

A black hat hacker is a person who attempts to find computer security vulnerabilities and exploit them for personal financial gain or other malicious reasons.

(Quoted from Technopedia)

White Hat Hacker -

A white hat hacker is a computer security specialist who breaks into protected systems and networks to test and asses their security. White hat hackers use their skills to improve security by exposing vulnerabilities before malicious hackers (known as black hat hackers) can detect and exploit them.

(Quoted from Technopedia)

Backdoor -

A backdoor is a technique in which a system security mechanism is bypassed undetectably to access a computer or its data.

(Quoted from Technopedia)

Ponies don't have hands

This has never been a problem ever

5703221 I did have someone tell me that due to them not having hands it would be difficult for them to use a mouse. Therefore, I posted this here just as a means to address the issue.

5702754
Most programmers would actually use this definition:

Playful solving of technical work that requires deep understanding, especially of a computer system.

And refer to what you're describing as "cracking".

5704747 True but there is a difference between hacking and cracking. The way it sounds someone who is cracking may have permission to mess with the programing while an individual who is hacking generally isn't.

At least I think that is the case.

5705784
Cracking is about breaking into systems. (Think about "safe cracking" — same thing with computers instead of safes.) This may be with permission of the owner to test their security, or may be without permission for various other reasons.

Hacking is about coming up with creative uses for technology to accomplish your goals. Now, those goals might be cracking, but might also be something more constructive. For example, setting up your tablet to take a picture once per minute and post them on Instagram, pointing it at your oven while your dinner is cooking, then going down to a bar to wait for your food to be ready, checking on its progress on your smartphone… that would be an awesome hack. But it's not cracking.

5705879 Thank you for pointing this out to me. I had to do some research on the matter (merely to clarify the subject) and have made some changes to the original thread.

Depending on opinions I might actually take out the hacker term definitions I found and give them their own sticky thread. We'll see.