If you are using a standard lightsaber, you may not have any animations. This is because you are missing a form with a stance. A lightsaber can have a set of forms that are given to you and cannot be added to, or are automatically given based on your Job or Usergroup.
If you have a personal lightsaber, you need to unlock these from Skill Trees. To unlock a Form or Stance, you navigate to the Character Station. By default, you begin with a FORMS tree.
Once you are in the Character Station, click on Skill Holocron, then click on View Skill Trees.
Navigate to the tree of your choice, click to open, and unlock the skill you want. (Provided you have enough skill points that is!)
To open up the Form Selection Menu, press your USE key (default E) and your RELOAD key (default R).
To change your stance, press your USE key (default E) and PRIMARY FIRE (default LEFT CLICK). It will cycle through the stances.
To use a Heavy Attack, hold your USE key (default E) and SECONDARY FIRE (default RIGHT CLICK). It will perform the heavy attack of the FORM you have selected.