So you have a guitar and an amp and you want to record it.
You have two main options:

Use a mic to capture the sound through the speaker
Use the line out connection of the amp, if available

There is a third option these days, but I'll leave that for later.

Using a microphone

What do I need:

Microphone: preferably one suited to recording guitar. One of the standards for this purpose is the Shure SM-57, a cardioid dynamic microphone tuned for the clean reproduction of amplified and acoustic instruments. You can also use a condenser mic if that is all you have.
Audio Interface with a mic pre-amp.

Connection set-up:

Connect the guitar to the amp as normal, and set up your sound.
Connect the mic to the interface.
Set-up the level for the mic, remembering to never let it go into the red.
Mic placement is a very personal and tricky matter, I suggest starting with placing it in directly in front of the speaker.

Pros & Cons:

Pros

You can capture the full sound you hear coming out of the amp.
No need to use any other equipment or effects to get a credible guitar tone.

Cons

Mic placement can be a long process, and not very easy to replicate if you have to disconnect, or move, the equipment.
It's complicated to isolate the sound of the amplifier if it's all located in the same room. In a studio environment the guitar amps are set up in one room, with the monitoring being done on a separate room.
Not suitable for late night recording, at least if you have neighbours or family.

Using the line-out

What do I need:

Audio Interface

Connection set-up:

Connect the guitar to the amp as normal, and set up your sound.
Connect the amp's line-out to the interface.
Set-up the level for the input, remembering to never let it go into the red.

Pros & Cons:

Pros

Simple set-up, no need to fiddle about with mic placement, and you need less equipment.

Cons

The resulting tone can be less than ideal, as most amplifiers don't have a speaker simulation on the line-out.
Still not suitable for places or hours that require low levels of noise.

The third option

With todays technology there is a third path available for recording guitar with an amplifier...

Using an amplifier that uses emulation technology with a integrated audio interface.

This means that you only need your guitar, amp and a computer to record. Two cables, a instrument cable and a USB cable. And you can use the amp as a monitor.

I truly recommend having a look at Yamaha THR line of guitar amps for an affordable, versatile, portable and capable of great tones. It even comes with a copy of Cubase for free.
This is my current setup and it just does the job!!!!

Usefull Links:

Shure SM-57

Yamaha THR collection