What if our galaxy didn't have a SMBH?
The supermassive black hole (SMBH) in the center of the Milky Way (MW) — called Sgr A* [Sagittarius A-star] — has no direct impact on our galaxy. Its mass is only a few million Solar masses, and if you remove it$^\dagger$, it will only affect the most central stars, which would suddenly continue in straight paths out through the MW. These stars would almost surely not hit any other stars or something like that (since stars are really, really far apart), but some of them have velocities high enough that they may escape the MW.

If Sgr A* weren't there to begin with, things might look a little different. There seems to be a relation between the mass of a galaxy's SMBH and the velocity dispersion of the stars in its central bulge; the so-called M-sigma relation. so MW without Sgr A* would mean a more ordered center. Our Solar system is located in the disk, far from the center, and their is evidence that SMBHs have little impact on the disk (Gebhardt et al. 2001). However, in their early phase (as an active galactic nucleus), their extreme luminosities cause galactic superwinds which blow out gas and may quench star formation (Tombesi et al. 2015).

$^{^\dagger}$Removing Milky Way's SMBH is left as an exercise for the reader.