For a long time, web designers were (and sometimes still are) obsessed with making web pages appear the same in every browser. You can see where this idea comes from - afterall, with print design, designers have absolute control over the printed page.
But not only is it not pOSsible for a design to look identical in every browser - it's not even desirable even if it were pOSsible.
Firstly, browsers are technically called "user agents" - the important term here is "user".
The user is in control of many factors of their experience - the might zoom the font size up or down, they might increase or decrease the page size, and so on.
So, a page may look very different on the same system, with the same setup, and the same browser, depending on the user's preferences.
Next, we have the reality that even with completely bug free browsers, pages in different browsers on different platforms may render differently for a number of reasons.
The font smoothing and rendering algorithms on Windows, Mac OS and other operating systems are all different.
The native DPI of different operating systems is different
The physical display devices have different color temperatures, physical resolutions, and other physical characteristics.
And then of course, as web browsers appear on more and more devices - televisions, mobile phones, game machines, and so on, the form factors move further and further away from the classic 1024 x 768px XGA screen.
Then, we have the challenge that over time specifications like CSS, HTML, SVG change and improve, and newer browsers improve on older ones. They fix up bugs, and implement new features (for example shadows in CSS, the HTML5 video element and so on).
If our goal as web designers was to have the exact same rendering on any browser on any device, we'd
- expend an enormous amount of effort
- probably go insane
- be missing the point
The point is that it's not only OK for the same page to look different on different browsers, and for different users - it's healthy. It's how the web works.
So, instead of striving for designs which are identical in every browser, our goal should be to make designs that provide access to the information and services on a page, regardless of the browser or device a person uses, but which take advantage of the specific features of a device, or browser.
We might serve a totally different style sheet for a hand held device, or television based browser than for a high resolution laptop.
We might take advantage of newer CSS features in newer browsers, even though these aren't supported in many older browsers, provided that by doing so, we don't deny the users of the older browsers access to the services and information on the pages.
That's the real meaning of "cross browser", and should be an important goal of web developers.