This is a problem I have noticed for some time, I have IE, Opera and Firefox installed (latest versions, history off)
When searching for a specific name, let’s try this:
http://www.google.com/search?q=alphaville
The top site is different in Firefox:
IE = alphaville.de
Opera = alphaville.de
Firefox = ftalphaville.ft.com
Actually Firefox page 1 for this keyword is totally different than the others, why is that ?
Are you sure that you are no logged in to your Google account in Firefox (Google promotes visited sites)? Additionally you may has a different browser language.
Nope, not logged in and a same browser language.
Yes, it’s very strange…
Do you get the same results in the search I posted ?
I’m GNU/Linux user and currently I have installed two browsers Firefox and Chromium (Chrome equivalent) and in both the result is same. Please check the request headers (especially field Accept-Language), you could also try to clear the cookies.
Cookies are cleared, no difference, not sure how to check the request headers.
Which site do you get as number 1?
The first is the COM page, because my Accept-Language is EN.
To check headers you can use Firebug in FF, or dev tools for IE9 [key F12], and Inspect element/NET tab feature for Chrome. I never used Opera so you have to find out yourself how to check this.
I see that you’re from Denmark so probably some browsers use your system default (local) settings to conclude the headers.
Similar situation happens with systems and encoding, my default encoding on GNU/Linux is UTF-8 but Windows use Windows-1250 (if I remember well). No matter, that I’m using the same browser. Browsers just use system local settings.
I have check and did not get any difference in IE and Firefox.
Is there a location setting - as I have found if the browser has me tagged in a certain location then results vary on PC’s & browsers?
There are local settings like language, timezone, etc. some browsers may shares these setting with operating system.
The Google results may be vary in case of vary request headers not a browser, but yes request headers may be dependent on browser and platform.
edit
BTW:
I tried to request [I]http://www.google.com/search?q=alphaville[/I] with Accept-Language DE, and then alphaville.de is in the first place. There is no magic 
How do I check the request header in Firefox? I have Firebug installed…
Enable firebug, and go to the Net tab. Reload the page and go to the HTML (or ALL) sub-tab and then click plus sign next to GET http://www.google.com/search?q=alphaville in a request list. You will see two tables Response Headers and Request Headers (this is the one).
Hmmm, can’t find it…I will try again tonight, need to get to the bottom of this.
I noticed that also. I am using chrome and my office neighbor is using firefox. I told him to search for a word in google to show him something but the results were different from my search. So, yes, the results are different.