The two PCs you selected look pretty good for everyday computing and basic gaming (older games, emulators). However, I'm not so sure they're going to run Skyrim at decent settings. They don't feature dedicated graphics cards, thus the game would be running off of the CPU's less robust graphical capabilities.

You might be hard-pressed to find something around $500 that will be worthwhile, I'm afraid. Especially if you haven't factored in other potential expenditures such as a monitor, stereo speakers, software (Microsoft Office), etc.

Newegg has a gaming desktop section that you might want to peruse. You can still find some at reasonable prices, but you might want to check your needs assessment since the models vary widely.

Operating System: Most people are familiar with/prefer Windows 7, but some of the models come at Windows 8, so be aware. If you go with Windows 7, I don't believe you'll need to be concerned about the differences between Windows 7 Premium and Professional, so don't worry about it.

Storage: 500 GB to 1 TB (1,000 GB) capacity should be enough for most folks who are relatively organized. If you have a huge music/movie collection or if you never delete anything ever, you might need more. If a solid state drive (SSD) is offered, consider it as this is a tremendous boost in speed, but at a capacity cost.

RAM: Generally affects how efficient your computer is at running multiple programs at once. 6 to 8 GB is enough for most folks, anything above this is probably overkill.

Graphic Cards: Most modern graphics cards will outperform your CPU's capabilities.

So just using this Avatar Gaming I5-4525 as a baseline, for $779 you get Windows 7 64 bit; a 1 TB mechanical drive; 8 GB of RAM; a upper-mid level quad core CPU (very important for everything non-gaming); and an AMD Radeon R7 250 graphics card (see performance perspective below). It also features an ethernet jack (no wireless), a DVD reader/writer, bluetooth, a mouse, keyboard, Uncertain how many USB ports it has. Not too bad of a package, though I've never heard of this manufacturer.

Intel HD 2500 graphics (from your second link): 1.8
Intel HD 4400 graphics (from your first link): 2.6
AMD Radeon R7 250: 5.3

That machine is all-around better than the two that you linked to - from both a gaming and non-gaming standpoint - and I don't think the price point is that much more demanding. So if possible you might want to make that your threshold. Just mind any additional expenditures you'd need to make, though Black Friday is also approaching.

This Avatar Gaming i5-4576 at a similar $779 features a beefier graphics card (rated 8.3 so now you're moving into "Skyrim and more" territory) at the cost of some CPU power and Windows 8.1.

If you are deadset on buying a PC without a graphics card, it might still run Skyrim - but I would try researching this through Google. For example "Intel HD 4400 Skyrim performance" or something.

I hope this helps! I tried making comments based on standard computer needs, but if you have any other considerations or questions, feel free to ask.