Whatever you are recording for, the ultimate goal will be to eventually share it and play it out loud. The basic use of a speaker or headphones is to listen to your product, maybe right after you record, or to play it back as you are mixing.
This section is for information on the different kinds of speakers and headphones you might encounter and what you can use them for. For definitions, see speaker terminology.
You may have external speakers for your desktop, and your laptop/tablet/phone will already have built-in speakers for playback. When mixing, it is fine if this is all you have, you can still get good results. They may not produce the most balanced sound, and depending on the quality of them, they may enhance different elements of your mix or playback, or distort them.
Like computer speakers, you probably have easy access to a pair of regular earphones or earbuds, it is better to use these, especially if you know that they provide a more reliable sound than a regular speaker. They also allow you to listen closer, without the sonic distractions of the space that you are in.
Studio headphones are a good investment if you are going to be mixing regularly. They help you get an even more reliable and balanced sound than regular earphones/earbuds/headphones. They can plug directly into your computer or audio interface, for direct monitoring as you are recording, and may require an adapter.
Purchasing higher quality speakers will help with your mixing process, more than using regular computer speakers. Finding a good pair will help with getting a more accurate sound.
There are different kinds of amplifiers, from ones specific to instruments (guitar, bass) that have the ability to alter sound as it is projected, to general ones that are meant for amplifying an accurate sound. Depending on what you need it for, it may be best to go towards a multi-purpose amplifier that you can use for different occasions.
This site is only for suggestions and examples on recording equipment and software and suggestions on how to set up your recording equipment, not exactly what you should buy and not exactly how to set things up. Depending on your space, equipment, and live/studio setting, you will have to experiment for the set-up that will produce your desired results. Don't be afraid to experiment and go outside of the suggestions laid out in this site.