Binary stars are actually pretty common (about a third of the stars in our galaxy are in some kind of binary), although the very close pairs are a little less common.
For the majority of the lifetime of the stars in a binary, both stars just spin around each other, burning their own hydrogen and orbiting tranquilly. However, unless the two stars are exactly the same size when they’re formed (not generally the case), then one star will run out of hydrogen before the other, and will transform into a red giant star.
Red giants are pretty fuzzy stars, only loosely held together by their own gravity. For some scale, our own Sun will become a red giant in its future. It will expand from its current (quite large) radius of 695,500 kilometers to something 200 times that large. But it won’t have gained any material to grow that much bigger - it just spreads out what’s already there.
If the star is on its own, this newfound wispiness doesn’t change much. But if there’s another star nearby, the other star can begin to tug on the outer layers of the red giant, pulling some of the outer layers towards itself. (This is the same thing that the moon does to our oceans.) Because the red giant is so big, the outer layers are only weakly tied to the centre of the star. If the companion star in the binary is close enough or massive enough, it can begin to tear the outer layers of the red giant away, and pull them towards itself. This will change the shape of the red giant from a diffuse sphere into a diffuse teardrop, with the point of the teardrop facing the companion star.
What the companion star does with that surface material depends on what kind of star the companion is. By the time one star has aged its way to being a red giant, its companion is in one of two states, which will be dictated by how massive the companion was when it began its life.
If the companion started out with a lower mass than its red giant partner, it will still be burning hydrogen in its core when the red giant begins to form. As it siphons gas off of the red giant, this companion star will grow in mass. Depending on how rapidly the red giant is growing, and how quickly the companion is siphoning material, the lower mass star can actually grow so large that it becomes more massive than the red giant. After having so much of its material drawn away from itself and onto its neighbor, the red giant will slow its donation of mass. This can result in a fairly stable configuration - the red giant, having lost a good amount of its atmosphere to its neighbor, will continue to slowly bleed gas into its neighbor’s gravitational well, and the neighbor will continue burning hydrogen until it has exhausted its own resources.
The other option for these binaries is if the red giant was the smaller of the two stars when they started their lives. This means that the star now turning into a red giant took much longer to reach the end of its life than its neighbor. (The bigger your mass, the shorter your life, if you’re a star.) So the companion star has already gone through its death throes, and can be one of a number of interesting stellar remnants.
The main options for your stellar companion in this case are: a white dwarf, a neutron star, or a black hole.
If your red giant is pouring gas down onto a white dwarf, you will eventually trigger some kind of explosion: either a nova or a supernova. A nova is a thermonuclear detonation on the surface of a white dwarf, and can recur multiple times, as it’s just a surface explosion. This kind of behavior makes these binaries fairly noticeable, because the brightness of the star will flare to many times its original brightness. A supernova, on the other hand, will detonate the entire white dwarf, blasting itself apart, and leaving nothing behind (also quite noticeable). This kind of supernova occurs when the white dwarf gains too much material to be stable (these stars are balancing gravity against an electron’s unwillingness to be pushed too close to another electron), and some trigger in the core sparks a runaway burning of material.
If the other object is a black hole or a neutron star, you’ll wind up with what’s called an X-ray binary, for the somewhat boring reason that it produces a lot of X-rays. For these objects, as the gas from the red giant is pulled off of the red giant star, it gets pulled into a very thin disk, surrounding the black hole or neutron star. The disk forms because it’s very hard for gas to lose a lot of momentum all at once and plunge straight down onto the black hole, but as a result, the gas winds up heating up to an incredible temperature before it makes it all the way to the neutron star or black hole. This heat causes the X-ray glow, and keeps the disk itself almost invisible in optical light.
So there you have it! A binary system of stars with one red giant will result in the companion tearing the outer layers of the red giant star away. From there, you wind up growing the object nearby, or causing a nova, a supernova, or the creation of a lot of X-rays.
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