Compatibility with your router is a non-issue when looking into a wireless adapter. Whether your router is Wireless G, N or AC, most if not all wireless adapters are also backwards compatible. As for what the different numbers mean, they just reference the difference in wireless standards that improve (ie: go faster) with each iteration.
Right now, AC is the current fastest wireless bandwidth standard, operating at speeds of 1 gigabit per second and offering a much greater coverage area than its predecessor, Wireless N. Because of its speed, it is also referred to as Gigabit WiFi. Additionally, your wireless router will often broadcast two different wireless bands at different speeds: 2.5 GHz and 5.0 GHz. The faster band is usually meant for wireless adapters on desktop systems which will need greater speeds for video streaming, downloading, and online gaming.
Installation is pretty much painless these days, with the drivers coming pre-loaded on the devices. Simply plug the adapter into an available USB port on your PC and it will start automatically installing and searching for your wireless network. You can go simple and small with a nano wireless adapter from TP-Link, which can offer speeds of up to 150 Mbps in a package that is smaller than a US quarter.
If you’re looking to go all out, then Asus has some very powerful PCI wireless cards that mount into an available PCI slot on your motherboard and can transfer wireless data up to 1300 Mbps. The advantages of having a PCI wireless card is of course first for the speed, but having one is also convenient. You never have to worry about losing it when moving to a new place, and it also frees up another USB port in your system. Intel, Asus, and Edimax all make USB wireless adapters and PCI wireless cards. Find one that works with your motherboard and current system.
For something more midrange, look at one of the wireless adapters from D-Link or Rosewill, but other makers like TRENDnet, Netgear, and Linksys also offer adapters and routers at every price point. It just depends on your budget and use-case scenarios. You will see a lot of options even in the budget tier that offer “dual-band”, which just means that they can pick up both the 2.5 GHz and 5.0 GHz wireless bands that your router broadcasts. Rosewill and D-Link also offer PCI wireless cards, just at a more mid-level price range. You may not get the same speeds, but you will still free up an additional USB port. Some adapters are also able to transmit Bluetooth, which will allow you to connect your phone or mobile device to your desktop PC.
As more gadgets and everyday household appliances become “smart”, the more you have to think about the strain they are all putting on your wireless network. While it’s certainly impressive that a refrigerator can send automatic reminders that the eggs you bought three weeks ago are about to go bad, you don’t really want to prioritize that over your Netflix streaming. While most smart devices require minimal bandwidth, all that can add up. Factor in several smartphones, tablets, gaming consoles and gaming PCs, and you’re bound to experience sluggish speeds that effect your video streaming and gameplay. Luckily, a lot of wireless routers come with their own management software that lets you customize and prioritize traffic. Bear in mind also that even really powerful routers have their reach limits, so investing in something like a wireless bridge or hub from Netgear, TRENDnet or D-Link can really help clear up the dead zones in your house or apartment.