How to get the width / height of an element in Photoshop?

When you make a selection, the 'Info' panel should show the width (W:) and height (H:) of the selection. It has to be the active selection, though.

Note: The active selection "marching ants" don't show up in the screen capture.

If you activate the transform tool on a layer (CMD+T or CTRL+T), you'll see that in the top panel, the X/Y position is shown along-side the size, angle, etc.

The size is in Percent by default, but right-clicking the field gives you a selection of measurements to pick from, Pixels, in this case.

Press 'Command-T' to transform the current layer. That will also show it's width and height in the info panel, as per Robert's answer (but more convenient).