How to fix Macbook that did not completely install a new version of OS, so now its stuck in a error loop every time its turned on.
Your best bet is to just start over from scratch.


Hold down CommandR while powering on the Mac to get it into recovery mode.
Once the recovery mode menu comes up, choose disk utility, elect your Mac's hard drive and format it. If you have an SSD choose APFS, if not HFS+. If you are not sure, go with HFS+.
When that is done (we reformatted just to be safe) Quit out of Disk Utility and choose the option to install macOS.
You may need to either plug the Mac into your router with an Ethernet cable (fastest) or sign onto your WiFi network as the Mac will now download macOS.
Make sure it is plugged in and is getting power. Older (magsafe) power connectors) have lights on them. Green for charged, Orange for charging and let it do it's thing.
Follow the prompts to choose your language, etc. Note that your Mac will likely install the macOS it shipped with. You can upgrade to Mojave once you have a bootable Mac.
Now just let it do its thing. You will know it is done when it starts asking more questions about your User account, to sign into iCloud. Etc.


Note that in step to I have you reformat your drive. This assumes you already have a backup (Time Machine or other) to restore your data from. If not you have a whole 'nother set of problems. Let us know if that is the case.

I have you reformat your drive in case something (disk structure, files, etc.) is damaged. It's more of a precaution than anything else. you MAY be able to reinstall without this step. I would not do that unless I absolutely had to and had data on the disk I didn't have a copy of.

It would also be helpful if you included your Mac's model in questions like this. Depending on the age of the Mac some features are not available, like Recovery Mode...