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I recently began a class on nuclear rocket propulsion, and one of the first topics covered was various nuclear rocket cycles. I'll do my best to explain them using amazing MS Paint drawings and words. The first is the hot bleed cycle. In this cycle, some propellant does not go through the reactor, but is instead shunted off in a different direction. This is mixed with some of the propellant that has passed through the reactor, but not out the rocket nozzle, creating a relatively hot stream of propellant. This propellant is passed through a turbine, which then powers the fuel pump. After passing through the turbine, the propellant is exhausted overboard (on some designs this can be used for attitude control). Since the propellant that has passed through the turbine is at lower temperature than that which has passed through the reactor, some efficiency is lost. The NERVA design from the 1960s/1970s utilized the hot bleed cycle. The cold bleed cycle is similar, except no propellant from the reactor is used to power the turbine. As a result, the propellant passing through the turbine is colder, thereby reducing turbine efficiency. However, this does have the advantage of producing less thermal stress on the turbine components. However, since the mass flow through the turbine is larger, the cold bleed cycle is less efficient than the hot bleed cycle. The expander cycle cleverly avoids propellant wastage by passing all the propellant used in the turbine back into the reactor. This avoids expending propellant in the relatively low temperature turbine exhaust, and means that the expander cycle NTR has a higher specific impulse than the hot or cold bleed cycles.
