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Bitcoin mining with a CPU was once the norm, but these days it's pretty pointless as mining with a graphics card (GPU) is 50-100 times faster because they have a lot more of arithmetic processing units and they can do large amounts of bulky mathematical labor in a greater quantity than CPU's (further info about this can be found here).
Fortunately, while mining, a computer is perfectly usable: you can use your daily personal computer to full-time mine and you won’t notice any differences in your experience. The GPU is usually not used (below 15% of usage) under typical “office” kind of use (surfing the web, watching YouTube videos or movies) and dedicating the GPU power to mining won’t slow the machine down or lag it up on you. As long as you don’t start a modern 3D game you probably won’t notice any differences while working on the computer. You can even start or stop mining at any time without losing any of your progress, so part-time or “during sleep time” mining is profitable as well.
What you need is:

CPU: Any CPU made after 2005 is more than enough to run the proper software for mining. Having a greater CPU won’t give any huge performance impact on mining.

MOTHERBOARD: it must have at least one PCI-E slot. Some motherboards support SLI or CrossFire. In those, you can fit two graphics cards for mining, which doesn’t necessarily have to be interconnected with the bridge. It does not matter how fast the motherboard PCI-E slot is (16x, 8x, 4x, 2x, 1x) the performance will be the same. As long there is a PCI-E slot, you can fit a graphics card for mining.
If you want to build a system specially for mining you’ll start wondering how to fit more than 2 graphics cards in it. Well, graphics cards can also be inserted in the little PCI-E slots, but these slots have to be open on the right side (if not you can cut the slot border off to make it fit but you’ll definitely void your motherboard’s warranty) or you can buy PCI-E risers (flex ribbon cables to extend the slot) and adapters (to transform PCI to PCI-E) to fit 4, 5 or even 6 mining cards on a single motherboard.

MEMORY: 1GB is good enough. It does not matter what speed, timings it has or if it is DDR, DDR2 or DDR3 type of memory.

HARD DISK: any ATA SATA bigger than 30GB will do.

POWER SUPPLY: it should be a good quality power supply which meets the graphics card/s minimum requirements. The system will be under heavy load at all times, so be reasonable with it.

INTERNET CONNECTION: it does not matter how fast it is. Mining uses very low bandwidth (less than 10 KB per 5 seconds), it just has to be stable.

GRAPHICS CARD: as explained before, the graphics card is the most important factor for mining.
Among graphics cards, the most cost effective at the moment are the AMD Radeon HD 5XXX and HD 6XXX series. AMD GPUs are much faster than Nvidia GPUs in Bitcoin mining because AMD GPUs by design have more ALU’s (Arithmetic/Logic Units) which are the ones used in these types of calculations (a more detailed explanation can be found here)
The mining speed can be measured in Mhash/s, the faster the GPU makes these calculations, the sooner you earn the BTCs.
Here’s a performance comparison chart between graphics cards:

The numbers shown are not definitive, because the difficulty to get a block (thus gain Bitcoins) changes every 2016 blocks which is taking approximately 10 days and the Bitcoin value changes as well, however both the difficulty and the Bitcoin value have been stable lately.
As you can see, any of the AMD HD5XXX or HD6XXX graphic cards performs very well, being the HD5830 and HD5850 the best bang for the buck. Investing in one of them is a good choice. Remember, time is money: you’ll probably recover the invested money in less than two weeks, plus you get the graphics card! A little gaming here and there won’t hurt :).
Mining with your GPU won’t damage it or shorten its life at all. It’s important though to keep the card temperatures below 85ÂșC which are considered normal and operational, even under high loads (like when mining or playing a modern 3D game) graphics cards won’t exceed this limit with the default cooling they come with. If you want to monitor the temperatures of your card you can find a couple of programs that provide this data at the very end of this tutorial under USEFUL UTILITIES.
For an extensive list of graphics cards and their performance in Bitcoin mining, see the Mining Hardware Comparison.
Keep in mind that making your graphics card work all the time consumes a little bit of electricity as well, but it’s not significant unless you have a mining farm, in this case you should consider the electricity bill.
To calculate how many BTCs or dollars your card will make you can visit the Bitcoin Mining Calculator or even a more advanced calculator involving electricity and hardware cost: Bitcoin mining profitability calculator.
MONITOR: in order for the miner software to recognize the graphics cards, each card has to be connected to a monitor (unless you have bridged SLI or CrossFire configured). If you are planning to use just one graphics card then you’re probably good to go. But if you’ve got more than one, there are some monitors that have multiple inputs, this works as well. You could get away with it if you configured each worker to start working, one at a time… once the first one is running switch the cable to the next graphics card and repeat the process until they’re all running. Or you can even make cheap ”VGA dummies” to fool the graphics cards think there is a monitor there, but it’s simpler if you have as many monitor inputs as video cards.
If you want to become an expert miner you’ll want to know how to make a VGA dummy:
You just need three 50ohm ~ 150ohm resistors and a DVI to VGA adapter (which usually comes in the graphics card bundle).

Bend and cut the resistors’ wires and fit them into the adapter as shown in the diagram, then attach the adapter to the graphics card and restart your computer.