I am new to blender, and I would like to know, is there a way to change blender's default rotation mode? If there is, please let me know. Here is an example of what I mean. P.S I imported a .obj file.
Each of your object's origin points are at the center of the scene instead of the center of the object. In the pivot point mode you have selected (Median Point), objects will be transformed about the median pivot point of all the objects that are currently selected (in your case only one is selected). To fix the pivot point and have rotation occur as you want it to, go to Object > Transform > Origin to Geometry Or one of the other options if you prefer it to be at the center of mass or 3d cursor.
Your video does not entirely clarify (for me) what you are looking for, but I am guessing you are looking for the setting "pivot center for rotation/scaling".
Here's the blender manual link:
This is a good setting to know about. Setting the pivot center to "3D cursor" will enable the behavior in the first part of your video. The second part of your video uses a pivot center of "Active Element" or "Median Point". "Individual Origins" is useful if you're transforming several objects and you want to apply the transform to each individually, rather than everything as a group around a common point.
If there's some other aspect of the rotation you're talking about, you'll need to describe it more fully.