Markdown is used in Document Automation 
an example of popular markdown tool is Document Assembly. The way it works is that a document has a markdown wich makes some the elements machine readable. It is pretty simple, not more complex than editing a Wikipedia page. This simplicity is also wt Eris is aiming at with Dual Integration.
In simple terms it is a way or formatting a document in a way that it can be read by a human and by a machine. Wiki is using a markdown.
Eris Legal Markdown purpose is to ceate contracts which are human readable, machine readable and proven authentic against itself archived on a block chain.
What it does in simple terms: it is a system to bridge the gap between real world legal contract and smart contract. It is designed in a way that it can be used by lawyers, not only programmers. In super simple terms; For example, a normal contract involving 2 parties and a solicitor is authentic if a solicitor is authorised and keeps a copy in his archive. If solicitor is corrupt, so is a contract. If it is stored in a "distributed ledger" proved and maintained by the network rather than a central authority, it is more difficult to corrupt.
You can use ELM (Eris Legal Markdown) to write a contract, for example, Terms of Service or a non disclosure agreement, which is both a text contract and a smart contract. (this is in my own words, need to double check if I'm getting it right)
To install and use Eris Legal Markdown see this tutorial ]. Visit Legal Markdown on GitHub
"The Eris Legal Markdown system is the backbone of the Distributed application Software Stack’s legal compliance strategy." - K: Sounds great, but a bit heavy. I would like to phrase it in terms of what it does.