Rights - a security giving existing stockholders the entitlement to purchase new shares issued by the corporation at a predetermined price (normally at a discount to the current market price) in proportion to the number of shares already owned. Rights are issued only for a short period of time, after which they expire.
These are also known as ‘subscription rights’ or ‘share purchase rights’.

Warrants - a derivative security that gives the holder the right to purchase securities (usually equity) from the issuer at a specific price within a certain time frame. Warrants are often included in a new debt issue as a ‘sweetener’ to entice investors.