Generally speaking, bicyclists are required to adhere to and follow the same "rules of the road" as drivers of standard motor vehicles. This includes - but is not limited to - signaling lane changes and turns, giving others the right of way, obeying traffic signs and signals, etc.
Bicyclists are legally considered vehicle operators and, therefore, have the same rights to the roadway as motorists do. But with this right also comes the responsibility of obeying the same laws and regulations that motorists are required to obey.
More from the City of Seattle's bike code:
"Every person operating a bicycle upon a roadway, street, alley, or any other way open to the public is subject to the same rules, laws and duties applicable to the operator of a motor vehicle, except as to the special regulation of this chapter and except as to those provisions of the Traffic Code by which their nature can have no application."
Any law enforcement officer (city, county, state) is authorized to enforce all applicable traffic codes and other laws on bicyclists just as they do on motor vehicle operators (this includes the issuing of tickets).
And yes, bicyclists are issued tickets for running red lights. It is illegal for bicyclists to drive through an intersection controlled by red lights (traffic control signals) or stops signs. Again, all the same rules as motor vehicles apply -- operate your bicycle as if you were driving a car.