OWD - Stands for Organization Wide Default. These are the settings that apply to your entire org by default. These have to be defined for every object in your org. They can be:

Public Read ONLY - Anybody in your org. can read the records of this object.
Public Read Write - anybody in your org can read and write the records of this object.
Private - Only the owner of the record can modify the record.

Sharing: A mechanism used to grant access to the records of the object whose OWD is defined as Public Read Only or Private. It can be done via:

i. Creating Sharing Rules
ii. Manual Sharing - Sharing button available on all records. Only owner of a record can share the record at his own will.
iii. Apex: can be done through Apex code.

Permission Sets:

Permission Sets are similar to profile. They are used to extend the user permissions. For eg. If you have a Profile "A" which gives access to accounts, and contacts but not any custom object. There are several users belonging to the Profile "A". None of them should have access to other objects apart from Accoints and contacts. BUT of those users, you want that 1 or 2 users should also get access to other objects. SO you can create a permissions set in this case and assign those 1 or 2 users to this permission set. If there were no permission sets, you would have. To create another profile B with same permissions as Profile A plus provide permissions to other objects in Profile B.

I hope this answers your question.