What is the "Unconditioned" that the Buddha is talking about? And vinjana in Buddhsm In advaita vedanta, consciousness is the self. Consciousness has 2 modes: it can be pure consciousness, and it is described as "being aware of being aware" or the "I that I am knows that I am"; while the other mode of consciousness is consciousness entangled in objects, like feelings and thoughts.

Is pure consciousness, or being aware of being aware, the same as what the Buddha called the "unconditioned"? Or is the unconditioned the cessation of consciousness, as Buddhism sees consciousness as impermanent and not self?

They said if consciousness was not aware of itself, none of us would know that we are aware. I have this problem, my friend had the same, so we wanted to ask you if you could help us out.
OP: In advaita vedanta, consciousness is the self.


In Buddhism, it is considered all conditioned and unconditioned is not-self/non-self. "


  
    sabbe dhammā anattā
  


Therefore, there nothing that can be identified as a self. Not even Nirvana.


  OP: Is pure consciousness, or being aware of being aware, the same as what the Buddha called the "unconditioned"?


No. In Buddhism consciousness is just to knows what is felt.


  
    Then there remains only consciousness, purified and cleansed.What does one know [cognize]with that consciousness?
    
    One knows, „It is pleasant.‟
    
    One knows. „It is painful.‟
    
    One knows, „It is neutral.‟
  


Dhātu Vibhaṅga Sutta. See below for a full quote of the section on consciousness.

What is unconditioned in Nirvana.


  
    "There is, monks, an unborn — unbecome — unmade — unfabricated. If there were not that unborn — unbecome — unmade — unfabricated, there would not be the case that emancipation from the born — become — made — fabricated would be discerned. But precisely because there is an unborn — unbecome — unmade — unfabricated, emancipation from the born — become — made — fabricated is discerned."
    
    — Ud 8.3
  


Source: Nibbana


  OP: They said if consciousness was not aware of itself, none of us would know that we are aware.


In Buddhism, consciousness is part of the 6 sense bases. It knows what you have come in contact with. 


  
    On account of a contact that is felt ... conditioned by that contact ...
  


Dhātu Vibhaṅga Sutta. See the full section quoted below.


  
    (iv) The six groups of contacts are to be understood, thus it is said, and in what connection is this
    said?
    
    Dependent on eye and forms,
    eye-consciousness arises. When the three meet, there is contact.
    
    Dependent on ear and sounds,
    ear-consciousness arises. When the three meet, there is contact.
    
    ...
    
    The six groups of contacts are to be understood, thus it is said, and it is said in this connection.
  


Cha Chakka Sutta




  
    The consciousness element
    
    Then there remains only consciousness, purified and cleansed.What does one know [cognize]with that consciousness?
    
    One knows, „It is pleasant.‟
    
    One knows. „It is painful.‟
    
    One knows, „It is neutral.‟
    
    On account of a contact that is felt as pleasant, there arises a pleasant feeling.
    
    When one feels a pleasant feeling one understands, „I feel a pleasant feeling.‟
    One understands, „With the cessation of that same contact that is felt as pleasant,
    the pleasant feeling conditioned by that contact ceases, it is stilled.‟
    
    On account of a contact that is felt as painful, there arises a painful feeling.
    
    When one feels a painful feeling one understands, „I feel a painful feeling.‟
    
    One understands, „With the cessation of that same contact that is felt as painful, the painful
    feeling conditioned by that contact ceases, it is stilled.‟
    
    On account of a contact that is felt as neither pain nor pleasure, there arises a neutral feeling.
    
    When one feels a neutral feeling one understands, „I feel a neutral feeling.‟
    
    One understands, „With the cessation of that same contact that is felt as neither painful nor
    pleasant, the neutral feeling conditioned by that contact ceases, it is stilled.‟
  


Dhātu Vibhaṅga Sutta