Can I put the free VMware into prodution?

You can, but legally it is a license violation to do so.   As such, the answer is "no". As it is also a TOS violation for experts to advise how to get around licensing, then even detailing what you would lose by illegally using the software, then really not much i can or will add.

Buy a license.

Note there are virtualization product that cost less (Parallels's) for instance, or even freebies that run in LINUX that come with the O/S.   You can search for questions regarding suitability for open source virtualization products in EE and probably find answers you need about them.   They may very well be good enough for your needs.

VMware ESXi can be installed in a production environment as long as you agree to the terms and conditions as well as you are fine with the constraints (i.e. no vMotion, no HA, etc.).  If you want additional features then you purchase vCenter edition that can provide you the level of features/support you need.  Just so we are clear, there are 1000s of organizations using ESXi in production.

It depends on what VMware product you are talking about. The VMware Player (free) can't be used in a production environment (nor in a company), this is a version only for personal use.

But if it is VMware ESXi (free) you are talking about, then you can use it fully in a production or business environment.

and there are 1000s of Large Organisations using the FREE edition with no support as well!

I worked for VMware for 7 years and can confirm that Andy, Rindi and Mohammed are all correct. The answer is yes you can, but you do lose access to some of the important features and 3rd party software vendor support (like back ups).

I would advise on getting Essentials package as it would give you vcenter (you can use appliance and not have to pay for MS license) which would give you cloning, alerting, and APIs.  If don't have visibility at what you're infrastructure is doing you can be in some serious issues without knowing it.

You can also read this, which says nothing about License Violations...

http://searchservervirtualization.techtarget.com/tip/vSphere-and-vCenter-licensing-and-pricing-explained-a-VMWare-license-guide

.. IN all fairness, you only wrote "the free version" of VMWARE, which i interpreted as the VMPlayer version. This absolutely can not be used  in  a commercial/company environment.  My apologies for not being specific  in my response.

That's okay, mis-understooding, understood!!!

I read it as ESXi! (but then I'm a VMware Nerd!)