I'm completely new to Angular2 and TypeScript. I'm following Tutorial but I keep coping this error. Is it an error caused by the compiler or something?

It looks like you are using an older version of the compiler.
You need to download and install the TypeScript 1.5 beta (or newer) in order to use ES6 style imports and annotations.
You can check the version you currently have installed using:
tsc --version
It should give you Version 1.5.0-beta.
If you already have it installed, perhaps your PATH variable still points to the old folder, so edit your PATH variable to point to the latest folder, i.e.
C:\Program Files (x86)\Microsoft SDKs\TypeScript\1.5\;
Since you didn't accept the answer, I assume you are on OSx?
I had 2 instances of TS installed on my machine, I needed to point to the correct/newer one. Try this:
In Webstorm goto Preferences (Command + ,) -> Languages -> Typescript.
Command Line Options:
-m amd -t ES5
Compiler Versions:
/usr/local/lib/node_modules/typescript/lib
This way it will grab your global typescript instead of the one Webstorm may have set to default for you.