I'm new to instruments and recording. I have an electric guitar and I'm learning to use it. I'm interested in recording music, and I understand that I should use an audio interface. If an audio interface basically functions as an external sound card, do I not need a sound card in addition to it, or would my motherboard's built-in sound card be just fine along with the interface? Can I plug my guitar directly into the audio interface or will I need to plug it into an amp first?
Every modern computer comes with an on-board 'sound card' - the traditional name for an audio interface. Back in the day it used to be an actual plug-in card, typically the ubiquitous 'Soundblaster'. Now it's more likely to be a single chip, showing up in Device Manager as something like 'Realtec High-Definition Audio' (other brands are available). Though the onboard speakers (if any) will be severely limited by size, playback quality into headphones or an external speaker system will generally be very adequate.
This onboard 'sound card' will have a microphone input, but it's designed for the sort of mic on a headset that you'd use for Skype calls etc. There may also be a Line In function, which will be OK for direct connection of e.g. a CD player - or the Line Out of a guitar amplifier. If your guitar amp has a Line Out, and your computer has a Line in (more likely on a desktop than laptop machine), give it a try! What it WON'T have is an input suitable for directly connecting an electric guitar. This is not just about the type of connector, it's about the need for a high-impedence low-level input.
External 'sound cards' have Line In, Mic In (for recording-quality microphones) and MAYBE a special input for electric guitar. They don't honestly sound a lot different to onboard computer sound for playback - today's 'utility quality' is actually pretty darn good! But they do score on recording facilities and quality.
To return to your original question - an external 'sound card' replaces the function of the internal 'sound card'.
Free and up and running in 30 minutes, you will get a sense of the workflow and hopefully identify the parts you want to invest in first:
line-in, asio4all, SimulAnalog, VSTHost or DAW program.
It is possible to get your computer set up without spending any money (or at least very little), and then identify the parts you want to start upgrading after you see the basic workflow.
The built-in audio on your motherboard can be used as noted in the other answer. I originally used the line-in and a 1/4 inch to 1/8 inch adapter directly to the guitar.
The main problem to overcome with sound cards is latency. If the signal takes too long to be processed, you perceive the sound as being separate from the action of playing. Anything above about 10ms (milliseconds) is probably too much.
You will probably need to use an ASIO driver in order to get low latency. Some cards have one bundled with them, but for on-board audio, you probably will need the software package called ASIO4All (should be free). You might also want to set your audio chipset to use whatever sampling/specs that are native (e.g. 96khz instead of 44 or 48 etc) to minimize processing latency.
You will then want to use amp and pedal simulators. A common format for these is VST. A very decent set of free VSTs is SimulAnalog.
You will need a VST Host application to use these. A free one for Windows is called VSTHost.
Guitar Rig and Amplitube etc. are essentially locked-down VST host applications where you cannot add in your own VST plugins.
You can use the host to simply play music, but for recording you will want to use a DAW (Digital Audio Workstation) package. There are a number of free ones, and there is a decent try-ware one called Reaper DAW.
Most DAW software also acts as a VST host. What you do with the DAW is: add a track, set your guitar levels, add the VST sim(s) to the track, arm it for recording, and then record. The guitar is recorded as clean and "un-amplified." You can then alter all the amp and pedal effects to your hearts content without altering the performance.
You can attach VSTs to tracks in most DAWs and most stand-alone packages like Guitar Rig and Amplitube (mentioned above) can also be attached to a track.
One issue with ASIO is that it takes sole control of the audio while in use. This means other programs cannot use the audio subsystem. This is a good reason to migrate to a separate sound device for the recording stuff. If you do this, look for ones with proper hi-Z (guitar), mic, and balanced TRS inputs (i.e. made for recording use).
If you have a separate audio interface, you will probably want either two sets of speakers or a mixer so you can still use the on-board audio.
If you want to use an actual amp but use VST plugins, you will want some sort of re-amp device There is a do-it-yourself kit called L2A, but there are also off-the-shelf devices. These are for ground-lift and impedance matching.
There are a lot of questions here about ASIO, so do a music stack search for that. Hopefully, you have all the terms now to do a deep google/youtube dive for tips.
I listed a few Windows resources, but you can use the terms to find Linux or Mac resources as well.
In an audio chain, the weakest link determines the quality of the chain, so external soundcards do not utilize existing internal soundcards. There are setups like the by now somewhat ancient "RME DSP Hammerfall" where you have an external sound card talking via a proprietary digital interface card in your computer or laptop but those are not really in existence any more.
They have been superseded by Firewire soundcards and those, in turn, are nowadays replaced by USB soundcards (Firewire FW400 still had advantages over USB due to hubless symmetric design and bidirectional data transfer over the first batch of USB2.0 cards but nowadays USB has won out).
So external soundcards have for a long time been connected using general-purpose high-speed serial interfaces to computers and laptops alike, not involving any preexisting internal soundcards.
There is one exception in actual use and those are ADAT connections: in that case you use an external A/D-D/A converter (it's not called a soundcard any more) and connect it via "lightpipe" digitally to an external or internal "soundcard" that is no longer responsible for conversion but only for data shoveling and management.