Css – What does navbar-static-top do in Bootstrap 3

Short answer: no. (It is now possible with CSS transform. See the edit below)

Long answer: The problem with using "fixed" positioning is that it takes the element out of flow. thus it can't be re-positioned relative to its parent because it's as if it didn't have one. If, however, the container is of a fixed, known width, you can use something like:

#fixedContainer {
  position: fixed;
  width: 600px;
  height: 200px;
  left: 50%;
  top: 0%;
  margin-left: -300px; /*half the width*/
}

http://jsfiddle.net/HFjU6/1/

Edit (03/2015):

This is outdated information. It is now possible to center content of an dynamic size (horizontally and vertically) with the help of the magic of CSS3 transform. The same principle applies, but instead of using margin to offset your container, you can use translateX(-50%). This doesn't work with the above margin trick because you don't know how much to offset it unless the width is fixed and you can't use relative values (like 50%) because it will be relative to the parent and not the element it's applied to. transform behaves differently. Its values are relative to the element they are applied to. Thus, 50% for transform means half the width of the element, while 50% for margin is half of the parent's width. This is an IE9+ solution

Using similar code to the above example, I recreated the same scenario using completely dynamic width and height:

.fixedContainer {
    background-color:#ddd;
    position: fixed;
    padding: 2em;
    left: 50%;
    top: 0%;
    transform: translateX(-50%);
}

If you want it to be centered, you can do that too:

.fixedContainer {
    background-color:#ddd;
    position: fixed;
    padding: 2em;
    left: 50%;
    top: 50%;
    transform: translate(-50%, -50%);
}

Demos:

jsFiddle: Centered horizontally only
jsFiddle: Centered both horizontally and vertically
Original credit goes to user aaronk6 for pointing it out to me in this answer

To get the menu to automatically drop on hover then this can achieved using basic CSS. You need to work out the selector to the hidden menu option and then set it to display as block when the appropriate li tag is hovered over. Taking the example from the twitter bootstrap page, the selector would be as follows:

ul.nav li.dropdown:hover > ul.dropdown-menu {
    display: block;    
}

However, if you are using Bootstrap's responsive features, you will not want this functionality on a collapsed navbar (on smaller screens). To avoid this, wrap the code above in a media query:

@media (min-width: 979px) {
  ul.nav li.dropdown:hover > ul.dropdown-menu {
    display: block;
  }
}

To hide the arrow (caret) this is done in different ways depending on whether you are using Twitter Bootstrap version 2 and lower or version 3:

Bootstrap 3

To remove the caret in version 3 you just need to remove the HTML <b class="caret"></b> from the .dropdown-toggle anchor element:

<a class="dropdown-toggle" data-toggle="dropdown" href="#">
    Dropdown
    <b class="caret"></b>    <-- remove this line
</a>

Bootstrap 2 & lower

To remove the caret in version 2 you need a little more insight into CSS and I suggest looking at how the :after pseudo element works in more detail. To get you started on your way to understanding, to target and remove the arrows in the twitter bootstrap example, you would use the following CSS selector and code:

a.menu:after, .dropdown-toggle:after {
    content: none;
}

It will work in your favour if you look further into how these work and not just use the answers that I have given you.

Thanks to @CocaAkat for pointing out that we were missing the ">" child combinator to prevent sub menus being shown on the parent hover