For a quality home recording studio, you'll need a good microphone, a solid microphone stand, a pop filter, a shock mount for the microphone and all the electronics needed for recording your sound. To get even better quality, consider investing in soundproofing.
When selecting a microphone, steer clear of headphone microphones. These are not designed for quality sound but rather just for a person to be able to be understood. Instead, do some research into stand-alone microphones. Some brands you'll want to look for are Blue, Rode and Sure, though it also depends on the type of recording you intend on doing. Singing, for example, shouldn't necessarily be done on a microphone designed for podcasting.
Once you have your microphone selected, invest some research into gadgets that will improve its quality. A pop filter is a must, as it minimizes air movement, such as what you hear on the phone when someone uses the letters B and P.
Another gadget that will improve quality is a shock mount, which suspends the microphone, usually on springs or elastic bands. When a microphone is attached to solid objects like a desk, it not only records the desired audio but also the sound generated by the vibrations on the surface it's sitting on. The shock mount acts like a car's suspension, minimizing the vibration and, thus, providing better quality sound.
The electronics used to record the audio depend considerably on the kind of microphone used. There are three distinct kinds of outputs from a microphone: USB, XLR and 1/4-inch cabling. These will be unique to the microphone, and knowing them will be necessary before you purchase any pre-amps or mixers, as they often only allow one type in, without running into some serious expense.
Pre-amps and mixers manage the sound between the microphone and the recorder, which eliminates audio issues before editing. They connect to the recorder, which can vary as well. Audio experts are divided on what is best to use to record. Some swear by using stand-alone audio recorders, as they minimize the chance of a single technical problem causing a loss of the recording. Others prefer the more straight-forward option of recording directly onto a computer. This is a personal decision, though there are many comparisons between the two options on the Internet for recording artists to use for research.