Most of the pentium dells aren't upgradable to core2 duo cpus; I would post this question in one of the dell forums on their website. There's also a dell section on this website. If you want to change boards, use ebay or craigslist and go with another dell board so it works with your dell windows installation.
As stated above by the people who know what they are talking about, it is unlikely you can upgrade that CPU to anything other than another Pentium 4. It most certainly will NOT support an E8400. It may support a Pentium D but that is not much of an upgrade. Save your money for a new computer and don't waste it trying to upgrade obsolete tech.
Dell uses proprietary motherboards and I can't really find much documentation on what you have except it had an 945 Express chipset. You should post on the Dell forums as has already been suggested to find out if you can even upgrade to a Pentium D.
Obviously you want to upgrade your computer. Buying a used computer from ebay is not the best way to accomplish this. Of the 2 you listed the Acer is about 10 times faster than the one with the Pentium D. I would not buy anything that does not include a detailed description though and the Acer does not.
The computer you have now is so old you will not be able to reuse many or even any parts. I assume you bought Windows 7 since there is no way that came preinstalled. So you might be able to reuse your copy of Windows depending on the type you bought. I'm actually surprised Windows 7 even runs well on what you have now.
Buying a $50 computer on eBay to put a new processor in it is an awful idea. Just build one up from scratch. We'd be happy to help you with that, but you're wasting your time, and potentially your money, looking at this kind of garbage on eBay.
What is your gaming expectation and budget? These two questions needs to be answered to know if they are realistic. If you don't have $500 budget, I would think you are better off with consoles unless you're happy with playing older games.