Okay, so I'm a big AMD fan, especially when it comes to laptops, and have experience with different AMD chips (and some Intel).
The main determinants here are going to be price/budget and use/purpose.
Different chips are going to give you better performance depending on what you want your laptop for. I'm not going to assume gaming because people have other uses sometimes, but here's a general breakdown of where things go:
AMD is always going to be cheaper than Intel. They tend to be more power hungry and run a little hotter and the Intels tend to run less hot and consume less power. There is a caveat here in that if you are willing to go refurb/used you can sometimes get an older generation Intel for cheap. It's usually best suited for business notebooks because businesses cycle through their contracts every 3-4 years so there is a constant influx of "refurb" business notebooks that are technically just used but usually have nothing wrong with them. I recently picked up an HP Elitebook Refurb for $160 after rebates from Newegg which has a 3rd Generation i5 and I spent $20 to upgrade the RAM and $50 for an SSD and for $230 I have a laptop that runs super fast and meets all my business needs. That said, I wouldn't try to game on it at all.
So gaming is where the budget really makes a difference. Intel integrated graphics tend to suck. Now that's not always true and they seem to be getting better. One of my Steam friends recently bought an Intel laptop and has been testing the gaming on it and has done okay, but I think his is an i7. So with intel you either want a discrete graphics card (which makes it even more expensive) or an i7 (which again, increases expense). For gaming, AMD has better integrated graphics at a cheaper price so you can do some reasonable gaming for a lot less than it would cost for Intel.
And so this ties in with what I've already said, but the purpose or use of your notebook is going to make a big difference. Business, school, casual use - then go ahead and get an Intel, because even a 3rd or 4th generation one is going to serve you quite well for a cheap price and even an i3 will be fine for those uses (although a little slow).
Gaming and you need to go AMD if you have a limited budget. And here is where the FX and A series are distinguished:
FX is an older chip and technically can be more powerful than the A series. Difference is, the FX is just a chip and while there are some integrated graphics, they are not as good as the A Series. The A Series is a chipset with 12 cores where 4 cores are dedicated to the CPU and 8 cores are dedicated to the GPU. Meaning it is an integrated graphics chip, but it is specifically designed to be the most efficient and give you the maximum out of the integrated graphics. I'm a big fan of the series for that reason. A few years back I had an A8 laptop and I was running stuff like Sleeping Dogs when it came out at playable settings. We're not talking 60 FPS at Ultra, but we're talking 30 FPS at medium type quality. Basically, the equivalent of a console. I don't know how they do it, but the A series is just really good at gaming on a budget. And so the higher the number the more powerful in the series. A4 to A6 to A8 to A10 (and I think there might even be an A12). I'm not sure why the A8 tends to be better reviewed than the A10, but technically the A10 should be better.
Depending on what you're looking for you can get an A10 for under $300. I did so earlier this year for a gift laptop by using a coupon on the dell outlet to get a refurbed Dell Inspiron with A10 and 12GB of RAM for $277. You want to make sure you get a decent amount of RAM because these integrated GPUs pull from the system RAM since they have no memory of their own. So you want enough RAM to make sure you can power the integrated GPU and the rest of the system. I wouldn't go less than 8GBs. Now these Dell refurbs come with a full year of Dell warranty so it's like buying new in that you get their top notch customer service along with it. I've also seen some new Lenovo laptops in that price range with the A Series in it.
If you have any more questions, let me know, but hopefully this should get you headed in the right direction for your needs.