Let's just get straight to it! These are the main components you need to have a functional studio:
1. Microphones. The mic's single purpose is to convert the mechanical energy it receives from sound sources into electrical energy so that you can record the signal using analog or digital gear. There are two major types of mics: dynamic mics and condenser mics. The difference between the two is in the mechanism each mic uses to convert vibrations into electricity. Generally, condenser mics have a higher sensitivity than dynamic mics and are therefore often used when recording sound sources with delicate high frequency content, such as acoustic guitars.
2. Preamps. They function to strengthen the signal coming from the mic to a level high enough that is good for audio reproduction (from mic-level to line-level). These are the babies that can bring your signature sound into the song (amongst other later components in the signal chain). Every brand of pre-amp has their own coloring and characters. Even the low end budget ones can have their use when unique situations arise. Pre-amps to audio are like lenses to visual - they determine whether the image is sharp or blurry. Granted, pre-amps that give sharp definition transparent sounds are expensive, but an engineer always understands first that different situations call for different measures. Sometimes the lower end budget gear might just be what we're looking for.
3. Soundcards/converters. When we talk digital music production, this is the third device you need in your signal chain. The converter's task (sometimes called as soundcards) is to convert the electrical signal from the mic into digital bits (analog to digital conversion, or A/D). In a modest home studio, usually the converter acts both ways (A to D, and D to A), but in higher end studios the engineer may use separate devices for each conversion process to maintain a higher audio reproduction fidelity. You can use soundcards that are only converters, or use audio interfaces - converters and preamps bundled into one piece of hardware. Audio interfaces come in USB or FireWire form, each with their own necessities and advantages. It can't be said that one is better than the either - it depends on the use and specification of the product.
4. Computer (or hard disk recorder). Obviously, all those bits need to be stored somewhere - this is where your computer comes into play. Normally, being a home studio enthusiast starts out as a hobby. It's fortunate that almost any computer purchased during the Pentium IV era (sometimes even PIII) can be used as an adequate starting computer to produce music. This works by you recording the sounds into your hard disk as WAV or AIFF files. But as you develop your skills, you might want to upgrade to a computer that is specially designed and manufactured to be able to handle heavy use of audio data and editing. There is a debate on whether a PC or Mac is better for studio use, but the subject is completely dependent on the personal taste, preference, and commercial needs of the musician or engineer.
5. Monitor speakers. Not your average multimedia speakers you can get at the computer shop (although you can start out with those also). To produce acceptable quality music, investing in a good pair of monitors is a minimum. Monitors can range from a very modest US$300 a pair to thousands even tens of thousands for the higher end ones (used in the most professional studio environments). Good monitors allow you to listen to the music accurately, as it is, without introducing artifacts or coloring. The purpose of mixing is so that the song you produce sounds the same when played back using various playback systems (from boom boxes, to home theater sound systems). Using headphones is considered bad practice, although some engineers claim they can mix well using headphones only.
6. Room acoustics. Unfortunately, the majority of beginning home engineers don't understand the importance of room acoustics. Acoustics is to music like light is to photography - the manipulation of these elements separate the pros from the amateurs. A good studio acoustic design may cost as much as half the cost to build the studio, but the results are often worth the investment (assuming you've done your homework). Many engineers, even pro ones, invest substantially in better gear but without better acoustics. This is not wrong, it's just that the increase in quality won't be as good as we estimated at first if we don't also upgrade the acoustic quality of the room we produce in.
These are the 6 major components of a signal chain, the 6 major components of a home studio. You can start out real simple by just using the computer you already have, purchasing a low end budget mic, connect it to a converter, and start singing or whatever esoteric art you have in your vision. The key is: never let limitation stop you from making art!