Convertible bonds are bonds that are convertible into ordinary (also known as common) shares at a pre-specified price at the option of the investor; they may be handed over in return for equity in the issuer. Clearly the investor will only do so if it is profitable, i.e. if s/he is able to benefit from a rise in the share price [and more so than the rise in the bond price that he may be enjoying (which is the same as a decline in the bond rate)].
Exchangeable bonds are corporate bonds that are exchangeable for shares in a company other than the issuer of the bond. The investor has an option to undertake the exchange, but does not have an obligation to do so.