<p>Is there any type of software in which you can animate the way the .stl object will be printed? </p>

<p>I'm not talking about what the end result looks like. I'm talking about a tool which acts like it's printing the given object as an animation.</p>

<p>I know it somehow depends on your printer but is there anything I can use?</p>

You may wish to consider
Craftware
for your purposes. It's a free program in beta form that does provide a tool-path animation for printing the layers. It is not so much specific to a printer as it is configurable for your own requirements.
The video
shows what I believe you are seeking at about the two minute point.
Simplify3D also provides such information, but is not a free program.