Types (most common):

• 3528 SMD LED
• 5050 SMD LED
• 5630 SMD LED
• adressable LEDs (e.g. WS2812b)

Specs:

• Length (m)
• RGB, different types of white or Brightness
• LEDs per meter (normal 30 or 60)
• IP Rating (waterproof?)
• Energy Efficiency

Hardware:

• Power Supply
• Controller
• Remote 

• 3528 SMD LED: is no real RGB. Every single LED can only display one color. They usually dont get hot at all.

• 5050 SMD LED: RGB, is brighter than the 3528 and usually doesnt get hotter than 30-40 ° C

• 5630 SMD LED: RGB, extremely bright, can get very hot

• adressable LEDs (e.g. WS2812b): You can programm LEDs with special Controllers. Examples are on the bottom of this page

For beginners, I can recommend Complete Packages. (Outlet or USB powered, you can decide)

Only if you decide to install several LED-Strips at different places, it's worth buying seperated Hardware.

Normal Controllers use Infra Red or Radio Frequency signals to change the LEDs.

If you want to use your Smartphone (e.g. Siri), I can recommend you the Phillips Hue Strips 2nd Gen.. They are a little more expensive, but smart!

Depending on your Strip Type, you need to find the right Controller. There are single color, RGB and programmable Controllers, with different connections. 

RGB Strips do usually not give out a good white color.

If you don't need RGB, take a single colored Strip. 

Different types of white can keep you productive (cold white, a bit blue) or comfy in your living room (warm white, a bit yellow), etc. 

I am using this Philips "Bloom" behind my Television. It's a great alternative for backlighting in your living room.