Similar question answered by Sabiha Sahrish Bhatti in The Community, however I am copying her answer in here for your reference-
Rights Issues are issued to existing shareholders of a company when that company decides to raise more capital via issuing new shares. Existing shareholders are given the "right" to purchase new shares at a discounted price (generally discounted - not always); if they choose not to take this "right" they can instead sell the rights to purchase the shares on a free market to ensure that their net wealth is maintained (as the increase in supply effectively devalues each preexisting share).
Bonus issues are generally associated with an investor being issues with extra shares than what they paid for. This can be as means of maintaining net wealth also (redistribution from company held shares to shareholders etc). This is the issue of an actual share that can then be traded on an open market.