WiFi has had different "standards" since it was created. 802.11a was the first, introduced in 1999. It was limited to 1 antenna and one frequency (2.4 GHz). As technology got better, not to mention the chip and code in the router, they progressed through the alphabet. Most people's first router started with 802.11g since that was about when laptops and broadband became popular. The last generation was N, released in 2007 and replaced by AC in 2013 (but popular in devices built on 2014). N was also the first standard that allowed a second frequency, 5.0 GHz.
First of all, the different frequency (called bands, since it isn't an exact single frequency) make a big difference. All AC routers are dual band, meaning they have both. (technically it is a N antenna for 2.4 and AC antennas for 5.0, but that's semantics) 2.4 tends to be further range, but it is really common. Your microwave puts out radiation in the 2.4 GHz band. The higher band of 5.0 is less crowded and generally has better speeds. (unless you have no other WiFi around you, and you need the distance of 2.4 for a clear signal.)
So let's compare N to AC, since you aren't going to find G anywhere except your tech junk pile. N supports 4 antennas, and AC supports up to 8. Each antenna can connect to an antenna in your device making it 8x speed... but rarely does even a laptop have more than 2 antennas. But! AC supports something called "beamforming" which is is basically a targeted WiFi signal. Instead of your router putting out a uniform sphere of radio waves, the multi antennas of an AC router will triangulate and beam the signal directly to your device. This makes it much much faster and more stable. You don't get better range with AC, but it stays strong until the end.
So the numbers. The numbers are the bandwidth. An N300 is a 300 Mbps speed. An N900 is a 900 Mbps speed. The catch is that if it is dual band, that speed is split between the bands. So N900 dual band could be 450 & 450 Mbps on each band. That is further split between the antennas, and as you know from above not all devices can use all antennas.
As for the range, it is a little controversial. Technically higher numbers do NOT equal higher range. However! Stronger antennas that allow higher speeds will of course put out more power and give higher range. It also means the antenna can receive the signal from your device even if it is weak. And more antennas = better reception.
Long story short. There are a lot of numbers, and here are my suggestions on when it is worth paying for the better router:
N600 is generally the cheapest I recommend. It works for 2-3 devices and has a good range. At that low of devices the speed isn't the bottleneck. N300 is the absolute minimum, and is for 1 device.
N900 is a good upgrade. It works for 5-6 devices, and also supports dual band. You should get an AC900 if possible, but at this range it isn't a huge deal.
AC1750 is the premium router. Some brands dropped it for the AC1900 or AC2100, which is essentially the same. This has 3+ antennas and really leverages the AC technology. This is my router, and I constantly run 10+ devices on it. I highly recommend getting this power of router, amazing value for the price point.
High end has gotten to AC3100. 4-6 antennas, super high bandwidth, basically can run a dorm on the thing. You wouldn't upgrade this until the new router tech comes out lol.
So hope that helps!
For more reading: This article is a little outdated, but mostly accurate :http://kb.netgear.com/app/answers/detail/a_id/21918/~/what-does-it-mean-when-a-router-is-described-as-being-n150%2Fn300%2Fn600%2Fn750%2Fn900%3F