Different texts about my site in search results

If I search for "amd forum" on Google and look at my site "AMD Forum" in the search results it's always different texts that shows up as description, probably taken from the latest articles.

Is it not possible to have a description about my site that always is the same? Other sites have that. Why not mine?

Google uses 3 different description in the SERPs. 1) ODP description 2) your own description 3) Description snipped from the content. The reason for that to choose best CTR or best description to show searchers. You can add special meta to remove ODP description but it's not here the case.
You have only "<meta name="description" content="Hemsida för AMD Forum" />" this as description which is against google guidelines (too short). You should make it a bit longer charater count at least 100 chars so google will not try to fill the rest area. Just make sure it's longer and it'll be fixed by itself.

Where do I change the meta name description?

You can change it at your admin cp Settings > Options > General Settings

Yes, I have "Sveriges största forum inriktat på AMD" there, so how can it be "<meta name="description" content="Hemsida för AMD Forum" />" then?

It should be more content there within the box where it writes "Hemsida för AMD Forum" only 21 chars. You need to add text data there. For example you can put there. "Sveriges största forum inriktat på AMD, Hemsida för AMD Forum..." (please make sure it's with out " "s) I think you should make abit longer and nicer description. I just dont know swedish language

I added this text instead:

"AMD Forum grundades 2008 och är idag Sveriges största forum exklusivt inriktat på AMD. AMD Forum är inte bara ett forum utan en komplett hemsida för datorentusiaster som föredrar AMD i sina datorer."

Please open your source code at browser and check if your change included. Because i did not see at codes it changed. It maybe a cacher issue needed to be cleaned.

Otherwise your problem is more specific and please check this thread Please help me change my meta description

I see there is a setting in vBSEO cp under "SEO functions" called "Replace the 'META DESCRIPTION' Content?" What is best for SEO? Should this be enabled or disabled?

My source code displays this:

<meta name="generator" content="vBulletin 4.1.0" /> <meta name="keywords" content="hemsida,amd,forum,index,artiklar,nyheter" /> <meta name="description" content="Hemsida för AMD Forum" />

It is NOT the same as I have set in the options.

It's for me off. But i use special plugin for translation. It does not allow it me to use it. Otherwise it would be on. Especially it helps to prepare more customised Meta Description for Member Profiles. You can trust vbseo standart settings.