Here’s an artist’s rendition of Arcturus (from Wikipedia):
Or, in comparison with some other stars:
So, a red giant is red, and giant.
Well, more a reddish orange than red.
In fact, they range in color from yellowish to more red than Arcturus, but still not fire-engine red.
But they are pretty giant. Still, seen from Earth, with the naked eye or a small telescope, they look just like a dot. But when you zoom in enough, well, you can see from the picture, even if it’s an artist’s rendering.
Even though they’re usually only about a tenth as bright as when they were in their youth, all that surface area adds up. That’s why many of the most well-known stars in the sky—Arcturus, Aldebaran, Gamma Crucis—are red giants.
You may also notice that Pollux and Arcturus are a lot spottier than the Sun and Sirius. Here’s a GIF (I couldn’t find the source) showing how the spots move around on Arcturus—like the tinier ones on the Sun, but more dramatic:
That’s because their convection cells are huge, even compared to the hugeness of the star itself.
There are also red supergiants, which are a different kind of thing. But, as you can guess, they’re even bigger. Still too small and far away to form anything more than a point on a home telescope, but here’s a comparison shot (from Wikipedia again). Look at box 5: Antares and Betelgeuse are red supergiants; Aldebaran is a red giant.