Using only CSS, show div on hover over <a>

You can do something like this:

div {
    display: none;
}
    
a:hover + div {
    display: block;
}

<a>Hover over me!</a>
<div>Stuff shown on hover</div>

This uses the adjacent sibling selector, and is the basis of the suckerfish dropdown menu.

HTML5 allows anchor elements to wrap almost anything, so in that case the div element can be made a child of the anchor. Otherwise the principle is the same - use the :hover pseudo-class to change the display property of another element.

.showme {
  display: none;
}

.showhim:hover .showme {
  display: block;
}

<div class="showhim">HOVER ME
  <div class="showme">hai</div>
</div>

jsfiddle

Since this answer is popular I think a small explanation is needed. Using this method when you hover on the internal element, it wont disappear. Because the .showme is inside .showhim it will not disappear when you move your mouse between the two lines of text (or whatever it is).

These are example of quirqs you need to take care of when implementing such behavior.

It all depends what you need this for. This method is better for a menu style scenario, while Yi Jiang's is better for tooltips.

I found using opacity is better, it allows you to add css3 transitions to make a nice finished hover effect. The transitions will just be dropped by older IE browsers, so it degrades gracefully to.

#stuff {
    opacity: 0.0;
    -webkit-transition: all 500ms ease-in-out;
    -moz-transition: all 500ms ease-in-out;
    -ms-transition: all 500ms ease-in-out;
    -o-transition: all 500ms ease-in-out;
    transition: all 500ms ease-in-out;
}
#hover {
    width:80px;
    height:20px;
    background-color:green;
    margin-bottom:15px;
}
#hover:hover + #stuff {
    opacity: 1.0;
}

<div id="hover">Hover</div>
<div id="stuff">stuff</div>