派出所：The grass-roots public security system,belong to the public security bureau，distributed in the communit,to maintain the community security. 公安局：A city can only have one,to maintain the public security of whole city.派出所 is an unit of the公安局。

派出所 refers to your everyday, local police station whereas 公安局 is a more official public security bureau (usually government related).

as we know, china has lots of people. for example, my hometown is a very very very small city, however, there is 300 000 people in my hometown...I have a french friend who came to my hometown and he said this was a very large city....:D so managing a city is a difficult work and is very different from western countries. china use a method which is: 派出所 + 公安局 in every city, there is only one 公安局 but there are many 派出所. one 派出所 is responsible for a district (every city is separated into some districts). in every 派出所, there is only serveral policemen, such as 8 or 9. and they have no weapon. when someone is robbed or someone needs help, and he calls 110 (in USA, it is 911). 110 belongs to 公安局. according to the situation(what happened, where is he etc), 公安局 will call the nearest 派出所 to the scene. then 派出所 will send some policemen by vehicle. after watching and knowing what happened, the policemen who are just sent will deal with the case and will call 公安局 to get some support if necessary. ofc, if a serious crime happens, 公安局will go to the scene immediately. so, 公安局 is just like the western police. they do the same thing. and 派出所 is the subordinate of 公安局.

共同点：都是土匪吃白饭的 不同点：公安局比派出所级别高

Jeyee's answer is perfect.