Contrary to what most people here are suggesting, I'd recommend you do use a plugin if you want to animate the move. Just animating scrollTop is not enough for a smooth user experience. See my answer here for the reasoning.
I have tried a number of plugins over the years, but eventually written one myself. You might want to give it a spin: jQuery.scrollable. Using that, the scroll action becomes
$container.scrollTo( targetPosition );
But that's not all. We need to fix the target position, too. The calculation you see in other answers,
$target.offset().top - $container.offset().top + $container.scrollTop()
mostly works but is not entirely correct. It doesn't handle the border of the scroll container properly. The target element is scrolled upwards too far, by the size of the border. Here is a demo.
Hence, a better way to calculate the target position is
var target = $target[0],
container = $container[0];
targetPosition = $container.scrollTop() + target.getBoundingClientRect().top - container.getBoundingClientRect().top - container.clientTop;
Again, have a look at the demo to see it in action.
For a function which returns the target position and works for both window and non-window scroll containers, feel free to use this gist. The comments in there explain how the position is calculated.
In the beginning, I have said it would be best to use a plugin for animated scrolling. You don't need a plugin, however, if you want to jump to the target without a transition. See the answer by @James for that, but make sure you calculate the target position correctly if there is a border around the container.