The i3 is an entry level processor, it has a base clock and 2 cores. The i5 has a base clock and a boost clock, it also has 2 cores. The i7 has a base clock, boost clock, but has 4 cores in most cases. The i7 is the most powerful out of the Intel processors, but also consumes the most power and can drain a battery like it's no one's business. The i5 can be a powerhouse but has limitations, and the i3 is not really meant for anything more than entertainment or light gaming. The i5 would be best if you're looking to do some gaming on a budget, if not try for the i7.
There are currently 4 generations of processors, the higher the better. Intel does a tick/tock upgrade pattern, one will boast a big upgrade, and the other a smaller upgrade. The 4th generation was a small upgrade and improves battery life, if you can I would highly recommend sticking with the 4th generation. You should be aware that there are not too many brands that have upgraded to 4th generation processors yet, most notably Samsung.
Lenovo is the leading brand in PC, they're the only mainstream brand that offers SLI or dual graphics cards in a laptop. Their U line and Z line aren't worth a breath but their Y line is by far the one best amongst the budget gaming laptops. For $800 you cannot beat the Y410p or Y510p.
Asus is a a great manufacturer but they refuse to do anything that actually makes sense. They have 1 multimedia laptop that can be use for medium gaming, and a couple gaming laptops that provide massive amounts of performance. They also have a very thin ultrabook that has Iris 5100 integrated graphics which are quite powerful. Everything else is focused on being thin and light but ignores the need for dedicated graphic and demands a premium price. They seem to not have everything together yet.
Acer is a good place to look for a budget laptop, but they insist on sacrificing one thing for the other. For example if you want something with a dedicated GPU and a 1080p screen you'll have to go with an i5-4200u, if you want the i7-4500u you'll have to
go with a 768p screen. They have no "no compromise" laptops.
Samsung was a good brand when it came to laptops, but now they have no clue what they're doing. They have not updated the rest of their laptops to have 4th generation processors, nor do they use Nvidia graphics anymore. If you want something with dedicated graphics you'll have to spend $1300. They are trying to focus on their absurdly thin laptops with QHD+ (3200x1800) screens which few people can afford at $1700+ and they have the audacity to put integrated graphics in. HD 4400 simply cannot handle games at over 1080p resolutions, it makes no sense. Samsung used to lead in battery life...now they call a 4 to 6 hours battery all day battery life. The only good thing about Samsung now is their aluminum chassis fetish, JBL speakers, and superior build quality.
HP used to hold most of the mobile computing market, then they turned to garbage...but now they're proving otherwise. My mom got a 15z-j000 which is a thin and light 15.6 in budget laptop. It has a 1080p screen AMD A8 processor, integrated graphics, (AMD has better integrated graphics) and a backlit keyboard...all for $600. It's a great multimedia laptop that has decent battery life and a nice cool feel, HP used to have heating problems but the cool sense feature seems to destroy that problem. HP doesn't offer too much at the moment in terms of multimedia laptops, but they do have some nice touchscreen budget laptops. Take a look at the 14z, go for the a6-5200 with the 8670m and 6gb or 8gb of RAM and you've got a laptop that will preform good.
Dell used to be a trusted brand...they they went public in the stock market and was the victim of investment vampires. There are companies out there that push for higher stock prices and when they're at their highest, they sell all the stock which is devastating for the company and can cause it it to go bankrupt. In the process the company puts out things that make the customers hate the company. They recently bought all of those stocks and are now a private company, if you wanted to go with this brand you'd have to wait for them to recover and put out some good products. They've already got some very high quality products out, but are still recovering.
Toshiba...avoid if you can. I have had 2 Toshiba laptops, they can take a beating and still work...but they fall apart VERY easily. The build quality is worse than garbage, at least for the laptops under $600. This laptop that I have now has amazing speakers and touch sensative buttons for Eco mode, WiFi, lights for the buttons, pause/play, mute, and volume up/down, but those are the only good things about it.
If you tell us your budget we can find you something that will work well.