Which modern distribuition of linux has minimal graphic requirements? My graphic card has only 64MB of memory.
The distro itself has very little to do with this, it depends on what programs you have installed, and they all have fairly low graphics requirements. First off, I'd recommend just trying a distro you're interested (Fedora, Debian, Ubuntu or OpenSUSE seem to be good choices for beginners), and see if it's usable with the default settings. 64 MB of memory on your video card is probably fine.
I believe none of those distros turn desktop effects on by default, but if things are running slow you can check. In Ubuntu, go to Go to the System menu, choose Prefences, then Desktop Effects. If "Enabled Desktop Effects" is checked, uncheck it. If you're using KDE, go to KDE Control Center -> Desktop -> Window Behavior -> Translucency and disable translucency and shadows.
If things are still running too slow, you might want to try a simpler desktop environment, like XFCE. To do that, you would just install the package via your distro's package manager, and then choose it on the login screen.