You question sounds like a simple one, but the answer is very big. Search these forums. There's lots of information here about how to get started. As a bare minimum guideline, you will need:
1 - your computer,
2 - headphones or earbuds,
3 - an interface
4 - a microphone.
5 - software
I like having a mixer as well, so that I can keep lots of things setup and ready to go at the push of a button, but you don't really need one if you have an interface. If you DO have a mixer, and your computer has a high quality sound card with stereo line inputs, the mixer and soundcard can take the place of the interface. That is my setup, but most folks nowadays use an interface. You need one or the other, not both. That is a mixer and a soundcard OR an interface. If you have neither, the interface route will be cheaper and simpler.
If you want to make high quality recordings you will eventually also need monitors, but if you just want to get started without spending a lot of money you can do everything with your headphones. Just remember you can't trust them to tell you what your mix is really going to sound like when you play it out loud on a sound system, so before you send round to friends and family, you'll want to export what you think is your final copy to an mp3 play or onto a cd and then go listen to it on a good sound system, or better yet several different sound systems, so you can go back and make adjustments as necessary. This is a tedious process which is why most folks here will rightly say quality monitors are necessary equipment. But they're a big expense and you can certainly get started without them. If you do decide to get monitors, do some research and be prepared to spend some money on them. Cheap ones aren't going to do you much good, because they won't tell you what your mix will sound like on real world sound systems any more than your headphones do.
As for software you can download a program called Reaper. There are lots of packages you can purchase as well, but Reaper works great.
A powered mixer has an amplifier built in so that it can drive speakers. A standard mixer only outputs line level signals which would go to an external amplifier which would drive the speakers. There are also powered monitors which have the amplifier built into the speaker cabinet.
Good luck!
J