How to create and save images that can be resized on Adobe Photoshop without getting pixelated?

I was wondering how do you create or save an image properly so that I'd be able to open it on Photoshop and resize the image without being pixelated.

I have been downloading free icons online and I always choose to download the PSD file. When I open the PSD file, it will a square shaped icon on the bottom right:

Our graphic artist (still a student and is currently on OJT) creates images using Adobe Illustrator, but when he saves the file as PSD and I open it using another computer, it won't have the square icon, so when I try resize the image (small to large) it will be blurred/pixelated, unlike when there's the square icon on the layer (apologies as we don't know what it's called).