A non-renounceable rights issue refers to an offer issued by a corporation to shareholders to purchase more shares of the company (usually at a discount). Unlike a renounceable rights, a non-renounceable right is not transferable, and therefore cannot be bought or sold.
Issuing more shares dilutes the value of outstanding stock. But because the rights issue allows the existing shareholders to buy the newly issued stock at a discount, they are compensated for the impending share dilution. The compensation the rights issue gives them is equivalent to the cost of share dilution.