Realistically, I don't think you can build your own laptop.

For a desktop, there are several components required. But what you need depends on what you start with. Assuming you already have a desktop computer, and you won't have any use for it after you build a new desktop computer, then you can probably use some of the components you already have in your existing computer. By the way, I have been building my own computers for years, and after my first one, I just upgrade one or two components every now and then (based on my current needs) rather than start from scratch. For example, my current computer is in the same computer case I bought over 10 years ago for my first build. Building computers this way makes them cheaper than what Dell or anyone else can do, so I always have a nice up-to-date computer but with only incremental costs when a good sale price comes along and my needs or desires require it. If environmental issues concern you, there's less waste this way, too.

When you buy components from newegg (or other places), some are sold as "retail" and some are sold as "OEM." (That will be right in the title of the item.) It's usually better to buy the "retail" version for most items. The difference usually is that retail comes in a pretty (decorated) box, probably has a significantly longer warranty, comes with all required accessories such as cables and fans (if appropriate), and comes with installation instructions. OEM components might not even be in a box and generally have absolutely nothing else but a 30-day warranty, and cost a small bit less. Especially for your first build, I recommend using only retail components. If you do buy an OEM component, you should have a good reason for doing so (and saving $5 is NOT a good reason). You get what you pay for, and retail is worth the little extra; OEM CPUs don't come with a fan, for instance, but retail CPUs do. If you ran your computer without a CPU fan, you'd destroy your CPU in minutes (and the warranty would be voided).

By far, the single most complex component, from a builder's perspective, is the motherboard. Virtually every component in your computer connects to the motherboard. The retail versions absolutely come with detailed drawings and descriptions, letting you know where to plug each type of cable and component in, and the types you need. Over the years there have been many improvements in computer components, and every now and then an advance requires abandoning the current standard for a new one. It's the motherboard that determines which generation of components you need.

However, probably the fundamental first decision to make is to choose which CPU you want. You can choose between AMD and Intel. Arguing which is better is like arguing which religion is best; they are both good, but they are not interchangeable, so they dictate which motherboards you can choose. If you want to be able to do incremental upgrades over the years to keep your system up-to-date but without having to replace other components, then choose reasonably current technology (doesn't need to be the bleeding edge, which has a large cost premium for a small increment in performance). You can get ideas of what to choose by first making a budget: about how much do you want to spend in total for your computer? Then review posts people have made that do the sorts of work you want to do, and see what they are using and whether they say it's adequate or not. Also, once you have a budget, go to newegg and look at the different components to see which ones are in your price range. That helps narrow things down a lot and give you a feel for what you should be considering. Newegg has customer reviews for each item they sell, so read those for an idea of quality and issues others have reported. You can read reviews of components at various websites, such as:
http://www.tomshardware.com
They have a lot of reviews and comparisons and so on which can be very helpful to give you the insight you might want to help decide what components to choose.

That's the background overview.

Here's a list of generally required components:
monitor
keyboard
mouse
speakers
computer case
power supply
motherboard
CPU
graphics card
memory
hard disk(s)
DVD or Blu-ray writer
an operating system (for Vegas users, that means a license to some version of Windows)

Depending on your setup, you might also want additional items like a wireless network adapter or a sound card (most motherboards have adequate sound capabilities built-in, but if you want to be doing recording studio quality work, then you'll want a higher-quality sound card).

You may be able to salvage a number of these from your current computer. If you buy retail versions (or can salvage from your current computer), then all the cables and adapters you need should come with them and not require any extra purchases.

There's no question that building a completely new computer from the ground up requires considerable research if you want to get everything that is compatible, low-cost, and high-performance, which might make it seem overwhelming. But future upgrades often involve just one or two items so it is usually much easier from then on.

For most performance computers, the main components to focus on are (1) the CPU, (2) the graphics adapter, (3) the memory (i.e., RAM), and (4) the hard disk(s).

For power supplies, they just need to provide enough power. The CPU and the graphics card are the main power users, so they pretty much dictate how much power your power supply must provide. You want to have excess to allow for future upgrades. Get at least 600W, but even more if you want a really powerful CPU or graphics card, or if you will be using several internal hard drives plus several internal DVD or Blu-ray drives. For many components, it is hard to find out how much power they require. It is better to have a power supply that can provide more than required than one that doesn't have the juice you need now or in the future. By the way, as time has gone on, newer CPUs and graphics cards have tended to require more power than older ones, so plan accordingly.

A final note (for this overview): most electronic components are very sensitive to static electricity, especially when they are not connected to anything that grounds them electrically (in other words, before they have been installed in a computer case). They come wrapped in special protective anti-electrostatic bags. You should do everything possible to protect them when you are working with them. Easiest (and most effective) is to only work on them when the relative humidity is high, i.e., not during the winter; if you still need a furnace on to keep warm, then your air is dry and static discharge is a serious risk. Do not remove a component from its anti-static bag until you are ready to install it; that means, do not walk between the time you take the component out of its bag and when you install it in the case (your feet rubbing the floor can generate a large static buildup). Avoid touching any subcomponents (the little do-dads soldered to boards); try to just handle boards by their edges and especially avoid touching any bare metal (other than the bare metal of your computer case which is good to touch so you are at the same potential as it). You can buy an inexpensive anti-static wrist band that you connect to your computer case so you are at the same electrical potential as it, which isn't a bad idea especially if you must work on your computer during the winter. Static electricity is a silent killer: a few kilovolts can shorted the life or outright destroy a component. For comparison, if there is a spark, such as when you touch a doorknob, then several tens of kilovolts have been discharged -- more than enough to utterly destroy any ungrounded component in its path.