Mixer:  A mixer combines a number of input sources allowing you to add effects and equalization before sending the total mix to one or more outputs.  A mixer may not be required if your soundcard supports multiple inputs and outputs, allowing you to connect multiple devices and route the outputs to different monitors or outboard gear.  A mixer can be very useful if your soundcard has a limited number of inputs as it will save you from connecting and disconnecting multiple cables!  Remember, however, that if you mix multiple signals before sending them to your soundcard, you won't be able to separate the signals afterwards.

Microphone:  a microphone will allow you to record audio into your soundcard.  The better the mic, the more accurate the recording will be.  Remember that your system is only as strong as the weakest link - a good mic requires a good soundcard and good monitors.

Pre-amp:  unless your soundcard or mixer has a pre-amp, you will have a hard time recording from line-level devices such as mics or guitars directly into your computer.  A pre-amp will boost the level, but will also "warm" the sound - an important factor when recording in the digital world.

Microphone:  a quality mic can make a huge difference in your recordings, so make sure you get the best mic you can afford.  Take note if the mic requires phantom power to ensure that your soundcard or mixer can support it.

Pre-amp:  consider if you need a multi-channel pre-amp for working with multiple simultaneous inputs, or if just a few channels will be enough.  You can also consider bypassing your soundcard altogether if the pre-amp supports a digital output or ...
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