I highly recommend Adobe Flash.  It is one of the simplest and most widely used animation programs, which means that it is relatively easy either to find animators or to learn the software yourself.  In the end, you have a product that looks something like the example in the first link.

If you want to also incorporate photographs or other raster images into your animation, you would also need a program like Adobe Photoshop or GIMP (which is free) in order to properly crop, scale, and otherwise manipulate the images. 

As a bonus, it's easy to turn a Flash animation into an interactive program.  The project I work for creates small educational demonstrations using Flash. You can see an example in the second link.

http://www.wiley.com/college/test/0471787159/biology_basics/animations/mitosis.swf

http://aspire.cosmic-ray.org/Labs/Machines/act2a/lab1.html

If you are looking for 3D animation I can recommend you Blender 3D at (blender.org). Specific add-on have been done for Blender:

http://www.autopack.org/home

To create 3D animation of cell biology behaviours, have a look on the gallery.

http://epmv.scripps.edu/about

To create 3D animation of molecules, have a look on the gallery.

http://www.bioblender.eu/

To create 3D animation of molecules, have a look on the gallery.

Hope it will help you.

Thank you very much Peter and Geoffrey for the valuable information. Really appreciated.

I would recommend Blender 3D which is an open source application. The software has user friendly interface to model and mold any 3 dimensional objects . You can also create industry standard animation with this application.

Inkscape for 2D and Blender for 3D pictures.

Blender for animations

For instance: https://www.youtube.com/watch?v=RasemoYzsFA

Perhaps you would like to try Molecular Flipbook. It's very simple to use and enough for most applications

https://www.molecularflipbook.org

What kind of software and How to create a 3D animation like this:https://www.youtube.com/watch?v=P5tuKJHsE-Y&index=1&list=PLx8NxTBoviX4VUz7Vd7lIyDN78KMqaCKs

This is a professional animation. I'd say various programs: Maya/Cinema 4D/MODO or some other 3D animation software, plus Adobe After Effects. Not for the faint of heart.

Thank you very much.

@ Peter Christiansen :

Do you mean Adobe Animate CC (Adobe Flash Professional) ??

Adobe Animate is a different option. 3D software is more realistic and Animate would be a cartoon.

3D builder is an easy option. One can fine model it in Maya or Blender. You can download free structures from many sites like GrabCAD, free3D.com, turbosquid.com etc and then manipulate as per your requirement. Please check the user rights/freedom after downloading.

Software name