A barbell is a piece of exercise equipment consisting of a long bar, usually with weights attached at each end. With a barbell and some quality weight plates, you can perform hundreds of exercises. Barbells are generally used to perform squats, bench press(flat or incline), deadlift, and barbell rows, but they can be used in any way seen fit. It is a great piece of equipment that are incredibly flexible in it's use.
A dumbell is a smaller version of a barbell that can be used individually or in pairs. Typically a set of dumbells can be used in the same way as a single barbell, but allows for more mobility and range of motion, as well as the choice to only use one out of the pair.
Machines usually consist of a stack of weights held to a cable by a pin. They can be configured in many ways to produce different results in different muscle groups. Machines are one of the easiest ways to learn what movements develop each muscle because almost all machines will have the name and a diagram printed on the side. They are also often overlooked by more experienced lifters because they believe that they do the work for them, although this is not true.
Cardio machines are a much different category from the rest. They are used to help raise your heart rate and help your cardiovascular system. This is mostly contributed to fat loss and heart health.