The Expressions Cheat Sheet

width and height are two reserved keywords in After-Effects that let you get the width and height of an item.

[thisComp.width, thisComp.height] // [1920, 1080] in a 1080p comp

In After-Effects, the top left corner of the comp is the zero point, where both X and Y are zero.
That means that the bottom right corner of the comp will be the point in which X and Y are equal to the comp dimensions: [width, height].
In a 1080p comp, this point will be set to [1920, 1080]. We can use "width" and "height" in expressions to dynamically get these values.

Naturally, to find the exact center of the comp we will have to divide both width and height in half.

Placing a layer exactly at the center

Try setting a layer's anchor point to its center, then apply this expression to the Position property:

[thisComp.width,thisComp.height] * 0.5;

That's right, dead center baby! Have you ever seen something placed truly in the center? But in the perfect, true center?

Now you can go to your comp's settings and change your comp size. Your layer's position will update to be exactly at the center using the new comp dimensions.

By the way, wondering what the square brackets are for? Look at your position property. It expects both an X and a Y (not to mention Z, but that's a story for another day...).
Your job is to give it back a value with a similar layout, both an X and a Y. [x, y].