Early jet engines were called turbojets where all flow went through the turbine. In other words there is no bypass. These are the cigar shaped engines you see in early airliners and military fighters. These types of engines are very loud and whiny and required high fuel flows to achieve the amount of thrust required.
Bypass allows a few things, it provides a cooling flow of air to mix with turbine exhaust and lower the overall exit temperature so you don't overheat your nozzle. The combination of the two exhausts can also see gains in thrust for given fuel flows. One of the most significant gains of bypass flow is noise reduction by having this outer flow "masking" the turbine noise. This is desirable in airliners.
There are two types of bypass:
Low bypass is where there is more turbine flow than bypass flow. This is common in business jets and military fighters with a ratio of around 1.5:1 or 2.0.
High bypass where there is more bypass flow than turbine flow. This is common in modern airlines with bypass ratios upward so 10.0:1.
In low bypass, the flow is remixed prior to exiting the nozzle, giving the cooler exit temperature typically to save the nozzle material. In a high bypass, the bypass air is allowed to flow outside of the engine after its high bypass compressor section, giving a very efficient airflow and adding to the overall thrust by moving a large volume of air outside the combustion and turbine section. This give greater fuel efficiency and much quieter engines for airplanes that can use them due to their size.
Below is an example of a low bypass F-100 engine which is found on F-15 and F-16 fighters.
This is the GEnx high bypass engine found on 747-8 and 787.