For Gaming GPU's, you will want to look at Geforce GTX or Radeon HD. Quadro or FirePro are Nvidia and AMD's line of workstation graphics (3D modeling and such) so they don't perform very well in games.
To answer your question about why lower numbered GPU models are higher up on the list -
For AMD and Nvidia GPU's, the first digit usually classifies the family they're in.
The second number usually classifies the sub-series in the performance line-up, the higher the number, the better.
i.e - GTX 280, Radeon HD 4890.
So knowing that, a GTX 280 will perform better than a GTX260, and a Radeon HD4890 will perform better than a Radeon HD4770.
It gets quite complicated but I hope it gives you an idea of what all the numbers mean. When in doubt, look at benchmarks, they're your best friend when you're just starting out They give you an idea of how much performance each card is capable of!
Also, just to clarify a few things, when you're looking at GPU's in a laptop today, you will most likely want to look for something that has the labels "GeForce GTX 6xx" or "Radeon HD 7xxx". The HD7000 series are the latest GPU's from AMD and the GeForce GTX 600 series are the latest GPU's from Nvidia.
Again, remember that the second number in the model name classifies the performance; the higher the better.
I just noticed I went on to ramble a little bit If I confused you in anyway, please let me know and I'll be more than happy to try and clarify things for you
It features the AMD A10 4600M (pretty decent IMO). It's a Quad-Core Trinity CPU with an integrated Radeon HD7660G GPU. This will most likely suffice for the max resolution this laptop supports (1366x768). Maxing out titles certainly wouldn't be in the picture, but playing modern titles will definitely be possible.
Generally speaking, the 5000 series and 6000 series really can't be compared head to head.
I'm not too sure about the 5870m and 6870m, however, I am well aware of the differences in the desktop versions of the two GPU's.
The Radeon HD5870 performs better than the Radeon HD6870, however, the Radeon HD6870 has better tessellation performance than the Radeon HD5870. The HD5870 may perform a little faster than the HD6870, however, in DX11 titles, the HD6870 generally tends to pull ahead by a small margin.
AMD/ATI changed the naming of their cards after the 5000 series so it can get quite confusing.
For example, in the 5000 series, there is a card called "Radeon HD5970". You would think that it would be the fastest single GPU card of that generation if you consider today, any card with the numbers "X970" are top-of-the-line (i.e HD6970, HD7970). However, that is not the case.
The 6000 series equivalent of the HD5970 would be the HD6990. They are both Dual GPU cards.
Long story short, the top-of-the-line single-GPU card in the 5000 series was the Radeon HD5870 while the top-of-the-line single-GPU card in the 6000+ series was the Radeon HD6970.
All-in-all, I would look at benchmarks of those two GPU's and comare performance. I will most likely end up choosing the HD6870m due to it being a newer GPU. I personally dislike getting something that's a few generations old