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I am trying to develop an application using Ethereum and it needs to have transactions which are not linkable.

I read an article by Vitalik Buterin, which speaks about it. He speaks about something called "internally contained private key" in the smart contract. What does he mean by that? Is there any way in which transactions to the smart contract could be encrypted?

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The short answer is no, this is not currently possible.

There are technologies that would allow anonymous transactions, such as zkSNARKs, ring signatures, and homomorphic encryption, many of which are described in the blog post you reference. Unfortunately, these systems are not quite ready for prime-time; for the most part they are too inefficient to work on the blockchain, although progress has been made by the ZCash team in incorporating zkSNARKs. See https://z.cash/blog/zksnarks-in-ethereum.html

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