When recording electric guitar there are two main aspects that make up the sound: The guitar and the amplifier.
There are also many smaller features that affect the sound as well. In terms of amps, there are sealed and open back cabinets. Open backs usually give a different characteristic to the low frequencies, in part because there's no air inside. They also have a tendency to fill the room with the sound, so you could argue that sealed amps are better in live situations.
There is also the option of valve or solid state amps. Effectively, solid state amps are more reliable, cheaper, and take up less room. So why are valves even still around? Most guitarists will suggest that you should always choose a valve amp over a solid state, because the distortion is very well rounded compared to the fuzzy clipping of solid state amps. Tone is everything and so valve amps aid guitarists in attaining rich, organic and yet overdriven texture and tone which sounds pleasing to the guitarists ear. It's no secret that the archaic musical methods such as unbalanced signals and analogue desks and recorders give a very vintage finish. Every decision you make towards your music purely depends upon personal preference. Some guitarists, particularly jazz performers, favour solid state amps for their 'cleaner', 'tighter' sound. It all depends on what you're going for and whatever happens to sound good! Moving on to the guitar, and a big part of the tonality is affected by the pickup. Again, there are two options for pickups: Single-coil and humbuckers. Single-coil was derived before the humbucker, which generally enhance the brightness and articulation of the the sound. Although the effect of a single-coil is more often preferred over humbuckers, single-coils are infamous for their hum, and as volume increases, so does the hum. Humbuckers, on the other hand, were designed with the intention of removing this. Rather than have one coil, they contain a second, which carries an inverted version of the noise, therefore cancelling it out from the signal. They tend to have a denser, more solid sound that can easily dominate in a mix.
After you've sorted the instrument equipment next comes the recording equipment. When choosing a microphone it's hard to make a wrong decision for electric guitars. Condensers and dynamics are both used widely for guitar recordings, though dynamics are used more commonly in the UK and condensers in the US. Condenser microphones are bound to give a bright, open sound where a dynamic would give a heavy, rounded sound. If you haven't been undertaking much experimentation as of yet it's strongly advised that this is where you do. Electric guitars are one of those instruments that has the ability to change in texture quite significantly therefore the right mic for the job will depend significantly on the overall sound of your guitar. Try out a bunch of different mics and simply choose which one you and your team believe to give the best finish to the sound.
Bear in mind that mic position has as much effect on the outcome as the mic selection does. While a large percentage of the sound is produced at the speaker there is also a lot of sound generated at the back and sides of the cabinet especially with open back amps. The typical way to mic up a guitar would be to position the mic in the centre of the speaker cone (where more than one cone is present, decide on whichever one sounds 'best' and go with that one). The centre gives a boost to the brightness nowhere else will, so if it's sounding a little sharp or ringy, try moving the mic around the speaker cone to find the nicest sound in terms of brightness. Try experimenting with different distances and angles, too.