You may have seen some rockets have giant "firey" looking exhaust coming from the engine with a lot of smoke, but you also might have seen other rockets with no smoke coming out of the engines, and just a little plume.
The first image is a Liquid Hydrogen and Liquid Oxygen fueled engine. The second one is a solid-rocket-fuel powered engine that releases a lot of energy by burning something. The reason some companies use mostly solid fuel over liquid fuel, like Arianespace and United Launch Alliance, is because solid fuel is MUCH cheaper than liquid fuel. Solid Rocket Motors are also much easier to operate, and simpler to create. Solid Rocket Motors can also provide more thrust, and can be ignited in less than a split second, which is why they are also used for abort towers, to quickly pull the capsule away from a failing rocket safely, efficiently, and quickly.
Some differences between liquid-fueled engines and solid-rocket-fuel is, you can shut down a liquid-fuel engine, but not a solid one. You also cannot throttle solid-fueled-motors, but you can throttle liquid ones. Liquid-fueled-engines do require a lot more of machinery than solid-rocket-motors. They are also very complicated and expensive.
Lot's of modern rockets like Vega, the Atlas V, the Ariane 5, and many more use solid fuel. Companies like Blue Origin, SpaceX, and more have opted to use liquid fuel for launching their rockets instead of solid fuel, as they could be potentially more dangerous. The Space Shuttle used Solid-Fuel for its two side-boosters, and the Challenger incident was caused by a solid-rocket side booster. Solid Rocket Motors are very good for launching rockets, but could be dangerous, and liquid-fueled rockets are usually safer, but are way more expensive and complicated.