Bonus shares means new shares given free of cost to all the existing shareholders of the company, in proportion to their holdings. For example, a company announcing bonus issue of 1:5, is issuing one (new) bonus share for every five shares held by the shareholders of the company.
Rights issues are a proportionate number of shares available to all the existing shareholders of the company, which can be bought at a given price (usually at a discount to current market price) for a fixed period of time. For example, a company announcing rights issue of 2:3 at Rs. 100 per share (current share price Rs. 130 per share), is issuing two (new) rights shares for every three shares held by the shareholders of the company at Rs. 100 per share. The rights shares can also be sold in the open market. If not subscribed to, the rights shares lapse on closure of the offer.