I really don't know whether this belongs in Laptop General Discussion, or this, but because it is specifically about the graphics. So, I have an old HP laptop with an HD 4000, and a 720p screen, and I somehow managed to get Titanfall at lowest settings at 20-30 FPS, with I somehow could play on. That laptop is no falling apart, so I'm looking at the Lenovo U31. I would like to know how much better the 920m would be better, considering the U31 is has a 15 watt processor, not a 35w (or 45w, not sure) 3230m, and it's a ultrabook form factor.
15 Watt vs 35 Watt CPU doesn't matter that much unless the 35W is a quad core. Intel bins the CPUs and charges more for the lower wattage versions. They're physically identical, the lower wattage CPU is just capable of running at a lower voltage without malfunctioning. The lower wattage CPUs tend to be clocked lower though, so you may want to compare clock speeds.
If you are planning to use the laptop for games, the 840/940m class cards are the lowest you should be looking at. Saving a bit of money over the course of 3-4-5 years you will have a laptop is not worth it at all if the trade off is not being able to run what you want or have to settle for lowest settings in everything. There is a great deal on a Dell system with a 960m card and a 256gig solid state hard drive http/www.amazon.com/Dell-Inspiron-i7559-763BLK-Full-HD-GeForce/dp/B015PYYDMQ
For something that is relatively inexpensive and provides decent gaming performance for the price (assuming you cannot afford $760 for the above linked Dell INspiron i7559), then consider the following Acer laptop with a Core i5-5200u and nVidia 940m with a 1080p screen for $490.
The 940m is not really powerful enough to play most modern games at 1080p unless they have low graphics requirements like League of Legends or Dota 2 so you are mostly limited to playing games at 1366x768 / 1600x900 resolution and low / medium graphic settings depending on the game.