Is there a way that works for all browsers?
Yes... one link here
window.screen.availHeight
window.screen.availWidth
Do you mean display resolution (eg 72 dots per inch) or pixel dimensions (browser window is currently 1000 x 800 pixels)?
Screen resolution enables you to know how thick a 10 pixel line will be in inches. Pixel dimensions tell you what percentage of the available screen height will be taken up by a 10 pixel wide horizontal line.
There's no way to know the display resolution just from Javascript since the computer itself usually doesn't know the actual dimensions of the screen, just the number of pixels. 72 dpi is the usual guess....
Note that there's a lot of confusion about display resolution, often people use the term instead of pixel resolution, but the two are quite different. See Wikipedia
Of course, you can also measure resolution in dots per cm. There is also the obscure subject of non-square dots. But I digress.