What factors should be taken into consideration by someone building a custom PC?

I've been a laptop-only user for almost a decade now, but I'm thinking of building myself a desktop soon so I can have more power. While doing some preliminary research, I came across the old-ish SU question Is it still cheaper to build your own PC?.

In the spirit of making questions less localized and more about general learning — per Jeff and others at my recent Meta question — here's a new question: What factors should be considered by someone building a custom PC? Just as importantly, what factors aren't that important?

As a subquestion, how do these factors change based on the intended use of the computer (e.g. software development vs. gaming vs. photo editing/graphic design, &c.)?

The single most important factor is compatibility.

It does you no good getting that super-fast DDR3 ram if the motherboard you selected only supports DDR2. Your new top-of-the-line Core i7 won't work so well if the motherboard is waiting on a firmware update before it can support it. Your new video card will be dead in the water if it requires a 6-pin power connector that's not included with your power supply, and your new ram upgrade from 2GB to 6GB will be a bust if you're running a 32bit operating system or you purchased mismatched ram from what you have that forces you to single-channel mode. There are potentially hundreds of issues like this in any given build, and you have to match up all of them to get the best system for your money.

Once you get that right you can start worrying about things like aesthetics, cost, noise level, etc.

The main driving force behind a building a custom PC is to achieve a hardware setup that is not available in the mainstream pre-built computer market.

Therefore, the intended use will be the greatest factor in the selection of hardware components. If you are building a database server, you might be putting in 32+ GBs of RAM and redundant hard drives. Where as if your use was a graphic design, you would be focusing on a multi-core processors and advanced video cards.

Once you have a goal it is just a matter of getting the best balance of hardware for your purpose yet within your budget.

If I were building a system today, three things I would consider:

1. Power - Will my chosen power supply be enough to drive the components of my system?
2. Quiet - Will the chosen components (drives and fans) be quiet enough for the location where the computer will be used?
3. Cooling - Will I be able to keep the system cool enough given the selected components and desired level of "quietness"?

Factor the value of your time into the cost. You may find that it's more cost effective to pay someone else to deal with the hardware integration issues.