Opacity of background-color, but not the text

Use rgba!

.alpha60 {
    /* Fallback for web browsers that don't support RGBa */
    background-color: rgb(0, 0, 0);
    /* RGBa with 0.6 opacity */
    background-color: rgba(0, 0, 0, 0.6);
    /* For IE 5.5 - 7*/
    filter:progid:DXImageTransform.Microsoft.gradient(startColorstr=#99000000, endColorstr=#99000000);
    /* For IE 8*/
    -ms-filter: "progid:DXImageTransform.Microsoft.gradient(startColorstr=#99000000, endColorstr=#99000000)";
}

In addition to this, you have to declare background: transparent for IE web browsers, preferably served via conditional comments or similar!

via http://robertnyman.com/2010/01/11/css-background-transparency-without-affecting-child-elements-through-rgba-and-filters/

I use an alpha-transparent PNG for that:

div.semi-transparent {
  background: url('semi-transparent.png');
}

For IE6, you'd need to use a PNG fix (1, 2), though.