In fission, energy is gained by splitting apart heavy atoms (uranium) into smaller atoms (such as iodine, caesium, strontium, xenon and barium, to name just a few) whereas fusion is combining light atoms (in current experiments two isotopes of hydrogen, deuterium and tritium), which form a heavier one (helium). Both reactions release energy which, in a power plant, would be used to boil water to drive a steam generator, thus producing electricity.
Fission is triggered by uranium absorbing a neutron, which renders the nucleus unstable. The result of the instability is the nucleus breaking up (in any one of many different ways), in the process producing more neutrons, which in turn hit more uranium atoms and make them unstable and so on. This chain reaction is the key to fission reactions, but it can lead to a runaway process, as in a nuclear bomb.
Fusion is a much harder reaction to achieve, however it yields more energy than fission. There is no chain reaction involved – hence there can not be an explosion – the reaction is achieved simply by getting the fuel hot enough and containing it tightly enough for the components to collide and fuse. The temperature has been achieved (over 100 million degrees), however confining the hot fuel plasma using powerful magnetic fields has taken a while to perfect. The behaviour of plasma is now well-understood, and so the building of a power reactor is simply a matter of overcoming engineering hurdles – work is expected to commence on the DEMO fusion reactor in 2030.