With Chromebooks, you will NOT be able to run standard PC games as it's running on Google's Chrome OS. HOWEVER, many Chromebooks have Android support. This will allow you to play Android games that your Chromebook's system specs supports (as Google checks this before allow you to purchase & install apps).
However, many Android games are typically better geared for tablet & mobile phones than a traditional keyboard & mouse... so you may not want to play such games on your Chromebook. Please be aware that Chromebooks also tend to have little storage space (16 -64 GB's flash-based storage) as the system is geared to be lightweight & online most of the time.
Please be aware that many stores have a VERY LIMITED RETURN POLICY (typically 15 days), so you will need to act fast! Additionally, laptops that are capable of playing most games (mid-range to high-end) tend to run around $800 - $1500. However, if you're willing to make considerable compromises (most casual & some mid-range), you could get away with a system for around $400 & up. You may want to talk to a store associate with considerable computer knowledge to get a reasonable recommendation (as a good associate will try to make recommendations based on your usage & desired budget).
Hope this clear things up!