Guitar Pickups are the circuitry required to take a purely acoustic instrument, and allow it to be amplified or recorded directly into your setup.
You can think of pickups as the translator between the physical motion of your guitar strings and the equivalent electrical signal that is required downstream.
Especially in the case of an electric guitar... the pickups are everything because the strings and body generate no useful amount of sound on their own.
So just to be clear, if you are planning on recording your guitar with anything other than an external microphone then you will need some type of pickup.
Also remember that you can record your pickups directly to one track and the acoustic sound to another as long as you have at least 2 preamps on your audio interface.
This can give you the best of both worlds when it comes to the mixing stage of your project because you'll be able to have 2 very different tones from 1 actual performance.
Guitar pickups come in many different brands and styles so don't be overwhelmed when trying to choose. They all fall into 3 categories:
The titles are pretty self-explanatory: some use mics mounted inside the guitar itself, some use magnets, and others sense vibration through a crystal (which is as magical as it sounds).
The type you choose will depend largely on what kind of recording you are trying to make.
Each has it's own strengths and weaknesses to consider before making a decision.
The following links can give you a much more in depth look for your preferred guitar style.