New to Arduino. I want to build a robot arm.

I want to build a robot arm as close to this as possible:

I have a 3D printer and will design and print all of the structural parts. As for the robotics, I'm fairly lost. I know along with an Arduino I'll need a few servo motors, but I'm not sure if any servo motor will work, or if there are certain kinds for different movements.

Where can I find a good place to learn about common parts used in Arduino projects, and what they're used for. That would help me learn a lot easier.

What have you done in the past with software and hardware?

http://www.robotshop.com/ca/en/arduino-kits.html

Nothing. I’ve been browsing that website all day but I have no idea which and how many parts I need off that site.

It's difficult to get to the sixth rung of the ladder without climbing up the first five. Why don't you back off, and start by buying a servo motor, progamming it to do a few things, and thus getting a sense of the real world aspects of motors, torque, current draw, etc..

Otherwise you'll order a bunch of parts, and find out half way through that they aren't the right parts.

It’s difficult to get to the sixth rung of the ladder without climbing up the first five. Why don’t you back off, and start by buying a servo motor, programming it to do a few things, and thus getting a sense of the real world aspects of motors, torque, current draw, etc…

Otherwise you’ll order a bunch of parts, and find out half way through that they aren’t the right parts.

I'm confident I can build what I want correctly if I can just figure out beforehand what each part does.

The building part is easy, the difficult part is making things work.
Suggest you get a starter Arduino, UNO, become proficient with writing code.
As suggested, learn to control servos.
Once you achieve these your project then can proceed.
It's my feeling the software to do similar to the YouTube will be quite advanced.

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If you get an Arduino Uno, and a servo motor, you can start playing with it right away, and it will help you plan the rest of the device. Saying that you are confident of your building abilities, while just a few posts prior, professing that you have not ever done any hardware, does not make sense. I have no doubt that you can figure it out, but physical building is a process. Unlike software, hardware has a physical reality that does things it's own way.

Go to the site 'Instructables' to find some examples of projects like you imagine. Don't try to follow one of their recipes exactly, they often have errors or oversights that can lead to frustration. But they do have a lot of ideas, and ... parts lists!