Liska,
Great question. First of all, try disabling all of your plugins and then try uploading an image. If you can, the turn your plugins on, one by one until you find which one is causing the trouble. In some cases, it’s not really the plugin, but your server doesn’t have enough resources allocated to it to run the plugin. That does happen on shared hosting plans rather often.
If you are still unable to upload an image after turning off your plugins, then try switching to the TwentyTwelve theme that’s bundled with WordPress 3.5. If that fixes it, let me know.
If you still can’t, most likely it’s an error with the server’s configuration. Either the sever doesn’t have enough memory allocated to it,or there is something amiss in a file called .htaccess that controls your server’s behavior.
It could also be that the files are your server are not set to be writable by WordPress. If that’s the case you can use your web host’s control panel file browser or a FTP program to set the wp-content/uploads folder to “755.” That should give WordPress permission if that was the trouble.
Let me know what you turn up and we’ll go from there.
Enjoy your week,
Bill