I use an HP G60 Notebook PC. I have 3 gigabytes of memory, a 250 gigabyte hard drive and an Intel Celeron processor. I bought it on sale at Walmart for about $250.00 about 3 years ago. I can run the game, but I do have some lag every now and then, especially on my Dynasty games where I have all of my store content installed. I created a different user name for challenge files that only has items from the base game, expansion packs, and stuff packs, with a few items from the store like clothing and such. I have less lag time on those game files, but the larger the house and the more sims in the house the more I lag. If you are eventually wanting to have all the expansion packs and stuff packs installed you really need at least 6 gigabytes of memory, 8 would be better, as LivvieLove has said. Anything less than that and you are pretty much guaranteed to lag to some degree.
My advice if you still aren't sure what you will need and want to keep the cost under a certain price point, would be to go into an electronics store like Best Buy, or someplace similar (or the computer guy that lives near you). Tell them you are looking to purchase a gaming computer, but don't want to spend more than what your limit is. They should know from that what the minimum specs are that you need. They can also give you expert advice on graphics cards and computer brands.