In this first post I'll try to provide some information on what methods that can be used.
Since this is a computer forum I'll focus most on the Interface with DAW method.

Question : WHY should I record my playing ?
Answer : There are many reasons like :
- It allows you to listen back to what you played so you can hear mistakes better compared to when you are busy playing.
- It allows you to let family/friends/other musicians listen and give their feedback (you can also post it on these forums or Facebook op YouTube for example).
- You are playing in a band and want to record your first album or maybe just a demo
- ...

Question : What recording method should I use ?
Answer : This depends on your personal interests and what you think fits your needs best.

Question : Then what methods are there ?
Answer : I'll talk about 3 methods :
1. Hardware Only (no PC/Mac needed).
2. Software Only (PC/Mac needed and some basic hardware).
3. Computer Hardware/Software combination (this requires an "Interface" and Software like a DAW).

Now let's get more detailed :

1. Hardware Only : This can be a simple Multi-Track recorder (like the TASCAM DP-004), these allow you to record from an input (guitar directly connected to the recorder) or to record by using it's integrated mic's.
You have these in a wide range of prices from cheap to expensive).
These are great for mobile recordings but lack most of the advanced features you have when using a DAW. However you can export the recorded song to a .WAV file and edit that on a PC (Downside is that in most cases the multi-track recorder will merge the various tracks on export so you cannot edit individual tracks, unless you export every track seperatly).

2. Software only : This is the most basic recording method. You just connect a microphone to your computers Mic In port and start recording with your favourite recording software (you can use the standard recorder that's included with Windows).
For this you only need a microphone that you can connect to your PC (some of the professional mic's might not work so you'll best bet will be getting one from a computer store). The quality of the recording may vary depending on your soundcard (integrated in your PC/MAC) and the quality of the microphone.
Like the Hardware recorders you cannot record individual tracks unless you record them seperatly, but it might be hard to get them all in Sync.
If you want to record for YouTube or Facebook you can also use a Webcam to record the video (I have a Loghtech HD webcam myself and the video quality is great but the build-in mic is not good for recording the audio part so I need to record the audio with another device and not the build-in mike of the webcam.

3. Computer Hardware an Software : This is what I personally prefer.
For this to work you need a special "Audio Interface" that can connect your BASS/GUITAR/MIC/... to your PC and which allows you to record in a DAW.
a D.A.W. is a Digital Audio Workstation (like Cubase, Protools, Garageband, Logic, ...)
Before you decide to purchase an interface you might want to do some research on the various devices and the bundled software.
For example : I use a Steinberg CI2+ (mady be Yamaha) this device came with Cuabase Artist 6. But a device by M-Audio will most likely come with ProTools (both are brands of AVID). And other might come with Ableton Live or something else.

So when purchasing you should find a best of both worlds (when I purchased my CI2+ package it came as a deal so I got Cubase almost for free, but this is no longer available and it now comes with Cubase 6 Elements and Wavelab 7 Elements (IIRC).

I would advise to look around for the software that you like best (I'm told that ProTools is the most popular one in the US while Cubase is the most popular in Europe, which for me is correct since I really like Cubase (and I've tried a lot of others).

When you've found the software you like you can look around for a deal with an interface and the DAW that you like (but it will in most cases be a Limited edition of the DAW, for example the cheaper CI1 interface comes with Sequel 2 LE and Wavelab 7 LE). If you cannot find this package you can still purchase the Hardware you like and get the DAW separately but that will increase the cost of your setup (and most DAW's are pretty expensive to but the full version so upgrading from a LE is a lot cheaper or when the LE does enough for you you don't need to upgrade at al).

Beware that some hardware might be specially build for a certain DAW, for example the CI2 serries has what Steinberg calls the AI Knob and Action Pad (for the CI2+ the CI2 has the AI Knob but not the fully featured Action Pad) this allows you to control specific portions of Cubase with the IA Knop and Action Pad so no need for a Mouse which might make it easier to use for some people.
This only works with Steinberg products like Cubase so the CI2+ as hardware combined with ProTools as software is not the best combination IMHO so you might be better off with an M-Audio interface or the cheaper CI1 that doesn't have the AI Knob).

Also some interfaces might support a Foot Switch that also allows you to do certain tasks without needing to use your mouse or keyboard (usefull if you are holding your bass in your hands).

DISCLAIMER : I'm not affiliated with any of the for-mentioned tools/companies. I mention some brands and products in my examples because those are the ones that I know but that doesn't mean that these are the best options out there. So my advise is to look around, get informed and use the method that you prefer with the hardware/software you prefer.