After attaining Nirvana, what happens to the person(conscious) after death?
Does the person cease to exist?
If not, does the person go to some peaceful place?
In one sense he continues just as we do. Once he attained nirvana under the Bodhi tree, the Buddha carried on teaching for another 50 years. It was only on the death of his physical body that he underwent parinirvana, that is to say final release. However this just pushes the question on from nirvana to parinirvana.
The question of what happens to an enlightened being after death is one of the unanswered questions i.e. does the Tathagata (Buddha) exist after death. The Buddha refused to address metaphysical issues such as these. He teaches one thing and that thing is liberation. He views other issues such as this, the nature of the universe, the nature of the 'soul' as a distraction.
It's not to say that this isn't a good question but it is one that is deliberately unanswered by the Buddha. As such I'm probably not going to be able to do it either.
It's the answer of no answer
This answer is based on Theravada teachings and is not the stance of all Buddhist schools. Other answers here will be from different perspectives and should be considered along with this answer.
There was no person existing in the 1st place to cease to exist. What you mistake as a person is just the 5 aggregates of clinging. They are just natural processes of causes and effects. Once you attain Nibbana, the causes for continuation of the 5 aggregates are removed. Hence no continuation.
As I understand it, Crab Bucket is correct that this is a question to which an answer is never given. I will contribute some source and try to flesh out this answer a bit:
Athma or soul gives life to the Body. A body is simply a tool for achieving the duties of an Athma, or Karma. Every time when someone or something is born, it is to do a set of duties. Salvation or nirvana is attained once and forever when a soul completes all its duties and free itself of its responsibilities. When a person fails to do its duties and roles, he is reborn and the circle of life, death and rebirth goes on till the soul achieves the same. The athma is assigned a new body and a new set of duties.
After attaining eternal salvation/nirvana the soul joins a circle of souls called pithru-athmas. Meaning athma of forefathers. The reason why the souls stand or sit in a circle is because among souls, nobody is greater or smaller, just like a circle has no end or beginning. Once the salvation is achieved the soul takes its rightfully earned position among the Pithru-athmas. Peace. :)
I'll take a stab at this one. The thing that needs to be understood for this is the idea of 'anatta'. This is not 'no-self' it is 'not-self'. The difference is this: 'No Self' is an opinion. A point of view. To know that there was 'no self' one would need to know everything at all times past, future and present. Holding points of view is thoroughly condemned by the Buddha as being merely opinion based on contact through the senses. http://obo.genaud.net/dhamma-vinaya/bd/dn/dn.01.6.olds.bd_end.htm The idea of 'not-self' is a thing which can be seen for one's self: "This" (and "That") are not self. The understanding that a thing is not self is based on the idea that if it were the self it could be controlled, such as, for example, to say that it was not to die. That cannot be, so that which is out of our control cannot be the self.
The dialog goes like this (from Vinaya Pitaka, MV 1):
"Consciousness is not self ... Inasmuch, monks, as consciousness is not self, therefore consciousness tends to sickness, and one does not get the chance to say in regard to consciousness, 'Let consciousness become such for me, let consciousness not become thus for me.' What do you think about this, monks? Is body permanent or impermanent?" "Impermanent, Lord." "But is that which is impermanent painful or pleasurable?" "Painful, Lord." "But is it fit to consider that which is impermanent, painful, of a nature to change, as 'This is mine, this am I, this is my self'?" "It is not Lord." ...
That understood it is to be extended to everything that has come into existence, understanding that 'that which has come into existence is subject to time, and subject to time, has a beginning, middle, and end. One of many ways Gotama used to describe 'everything in existence' was the khandhas (the stockpiles): body, sense-experience, perception, own-making and consciousness. Another was 'The All': the eye and sight, the ear and sound, the nose and scents, the tongue and tastes, the body and touch, and the mind and things.
What happens after death on attaining Nibbana (or Pari-nibbana) is best understood by first understanding how coming into existence occurs. That is the Paticca Samuppada, which is briefly: This being, that becomes, upon the ending of this, the ending of that.
Because the previously existing individual at death does not see the outcome in further pain and death in the assumption of existence, he acts in such a way as to create new existence. His identification with acts of body, speech and mind intended to create existence result in identified-with consequences taking the form of: contact with identified-with consciousness of named-shapes. "I see shapes."
If that individual had not acted in such a way as to create contact with identified-with consciousness of named-shapes, there would be no coming into identified-with existence and no coming to an end of that existence.
That is Nibbana. The sitting before, as it were, the possibility of existence and not doing that.
So it is not that 'a person' attains Nibbana, it is that 'the person,' that which has been previously identified with as 'Me' or 'my' or 'I', the khandhas, the all, is abandoned for the attaining of Nibbana.
Whatever it is that is left cannot be described because language (especially English with it's subject, verb, and object) is bound up in the terminology of existence. Nevertheless speaking conventionally there is mention of the Arahant experiencing un-identified with sense-experience (in his case the experience of sensations that are neither pleasant nor unpleasant), and consciousness (vinnana, the so-called vinnana anidassana, the unseen consciousness).
For a collection of materials on vinnana anidassana see: http://obo.genaud.net/backmatter/glossology/green_tea/green_tea_toc.htm