For computer recording, you need a source > mic > preamp > ad (analogue to digital) converter > interface > computer/software. Once recorded, you need DA convertion > monitors > working ears

. Everything depends on your budget and what you want from it.
Mics: For full band recording you will need some dynamic and condenser (require a voltage) microphones. People usually use dynamic microphones for close-miking drums, guitar amps, etc. and condensers for drum overheads, acoustic instruments, etc. Vocals are usually recorded with an LDC (large diaphragm condenser).
Preamps: These can be found stand-alone, in a mixing desk or in an interface. This boosts the signal to an acceptable level.
AD/DA conversion: These can be found stand-alone or in an interface. It converts the analogue signal to digital.
Interface: These take the digital signal into and out of the computer. A soundcard is a type of interface.
Software: There are so many options available, as with all the others, so it depends solely on what you need and your budget. Some basic softwares are free (audacity, etc) or come freely with interfaces (cubase le, etc.).
Monitors: You need to be able to listen back to the recording and mix, so you need monitors (speakers/headphones). It's important to have monitors which are as flat (in sound) as possible.
Acoustics: Acoustics are hugely important. If a recording is made in a small room, it will always sound like it's a small room and that is generally not a good thing. Each room has its own "sound" and acts like an EQ on the sound(with different settings for different parts of the room). You can change/influence the sound with acoustic treatment. This includes absorption and diffusion. It's generally a good idea to have a variable recording environment (able to change to suit) and a flat listening/mixing environment. For small rooms/studios absorption is generally recommend, especially bass traps in corners.
What i suggest depends solely on what you need and your budget. If you just need vocal recording, then you can get away with one or two tracks. For small bands recording individual, 8 tracks should be adequate. For a large band, recording simultaneously, more tracks will be required.
I usually, if the budget allows, suggest having more tracks than you think you will require. For vocals, you can usually get away with one, but I'll suggest 2, 4 or even 8 tracks to allow expansion. For a small band, recording individually, you can get away with 4 tracks, but i'd suggest 8+.
So, what is your budget and what do you need?