stars shine because of blackbody radiation - they just get hot enough to glow really brightly.
but for something to glow in x-rays, it has to be hotter.
like...a lot hotter. nuclear fusion isn't enough.
it may seem surprising at first, but matter falling and gaining kinetic energy is a more efficient way of heating things up than nuclear fusion by about a factor of 10. it's the same principle as hydroelectric powerplants, but scaled up to astronomical sizes. we call this accretion.
first, you have some gas lying about. then, you have some point of mass. that gas falls towards that mass and heats up, and that's accretion. usually the gas has some angular momentum, so it can't fall straight down, and instead spirals down in an accretion disk
Hubble image of an accretion disc near a young star, not that bright in x-rays btw
accretion happens pretty much everywhere where there's mass, but for the disk to shine in x-rays the accreting object must be SUPER FUCKING DENSE. oh hi black hole, oh hi neutron star...
at the largest scale you got quasars, a name which originally meant quasi-stellar objects, because from afar they look like stars. they're supermassive black holes in the very dense environments of the galactic centers, and accrete so much they're literally the brightest objects in the whole universe. active galactic nuclei are basically the same thing, terminology is confusing.
desy/science communication lab
at a smaller scale, you can get a stellar mass black hole or a neutron star in a close orbit with a regular star, and sucking up that shit like soda with a straw. they're called x-ray binaries (because they're binary stars bright in x-rays), and come in all sorts of shapes and sizes depending on the exact star and compact object present.
NASA
you can get even higher energy photons by scattering x-rays off really hot gas, or having some sort of particle accelerating shockwave present. the former happens a bunch in x-ray binaries, the latter in supernova explosions and astrophysical jets.