Open sidebar.php. You're looking for 2 lines. The first one (let's call it Line A) includes the code if (!function_exists('dynamic_sidebar'). The second (we'll call Line B) includes the code endif;. If you only have 1 of each, great - this should be relatively easy. If you have more than 1 of either, this could get a bit messy...
First, make a copy of sidebar.php in case everything goes pear-shaped.
Next, look at what lies between Line A and Line B. This is the code/markup that is automatically replaced by the addition of any widget. From what you describe, it sounds as if you have some code/markup outside of those two lines. If so, move this code so that it lies just before Line B (ie it's now between Line A and Line B).
Now save your amended sidebar.php file, add a widget to your sidebar and check the effect. If everything falls to pieces, revert back to that copy of sidebar.php that you made earlier and we'll try again.