The evolution of binary star systems is very complex because although the two stars may be coeval ( ie were formed at the same time out of the same collapsing gas cloud) their evolution rates are determined by their masses. If the two stars formed at exactly the same time, but one managed to accrete 3 solar masses while the other only accreted 1 solar mass, they will evolve very differently. The 1 solar mass star will evolve like our Sun, taking billions of years to burn its critical amount of core hydrogen, and after 13 billion years evolve into a red giant. The 3 solar mass star, however, will evolve into a red giant in only 3-4 billion years or so. After the high mass star has lost its outer layers and become a white dwarf, you will end up with a system much like Sirius with what appears to be an ordinary A-type star with a white dwarf companion. The white dwarf is all that remains of the more massive star which long ago ended its red giant stage. Sirius A, meanwhile, is still burning its hydrogen at a more leisurely rate and has another billion or so years to go before it takes its turn as a red giant.
Even more complex evolutionary scenarios occur when, in a compact binary system, one star evolves into a red giant and its companion begins to accrete an appreciable amount of the envelope of the red giant through a mechanism called 'Roche lobe overflow'. The star receiving the mass, then has its evolutionary clock reset to zero, and evolves as a more massive star. In extreme cases, the companion star may accrete enough mass that it becomes a supernova, leaving behind a neutron star, while the red giant which lost some of its mass continues to evolve into a white dwarf. Many of these neutron star-white dwarf binary stars are known. There are even some neutron star- neutron star systems such as the famous 'Binary Pulsar'.