Can't login because of kernel panic
This answer assumes:


your internal drive is full and this is why you can't log in
you do not have access to another Mac
you do not have another drive you can boot from.


The following steps will allow you to delete what we call a sleepimage file. These are usually quite large and should free up enough space for you to be able to log back in.

Note that while this can be done in much fewer steps (steps 4 to 10 could be removed and step 11 could contain a much longer file path), I've deliberately split this up to reduce the risk of making a typo and to make it easier to understand.

Deleting your sleepimage file via Single User Mode


Boot into Single User Mode by restarting your Mac and pressing and holding ⌘ + S as soon as you hear the startup chime
Keep these keys down until you see a black screen with white text 
Mount your drive by typing in exactly: /sbin/mount -uw / (note there is a space after the 't' and again after the 'w') and then pressing the enter key
Now list what's there by typing in ls and pressing the enter key
You should see a list of files and directories
Now type in cd private (note there is a space after 'cd') and press the enter key
Now type in cd var (note there is a space after 'cd') and press the enter key
Now type in cd vm (note there is a space after 'cd') and press the enter key
Now type in ls and press the enter key
Hopefully you will see that you have a sleepimage file located here (this is what we're going to remove) 
Now type in rm sleepimage (note there is a space after 'rm') and press the enter key
Now reboot your Mac by typing in reboot and then pressing the enter key


Hopefully this will free enough space for you to be able to login. If so, then take the time to remove any other data you don't need. If not, message me and we'll go from there.

NOTE: If your Mac doesn't reboot after Step 12, feel free to switch it off with the power button.

IMPORTANT: - You really should have a backup regime in place! Buy yourself an external USB hard drive and start using Time Machine for backups!


Deleting downloads via Single User Mode

Since you know you have items in your Downloads you can delete, these steps will help you do that from Single User Mode.


Follow Steps 1 to 5 from Deleting your sleepimage file via Single User Mode above
Now type in cd users (note there is a space after 'cd') and press the enter key
Now type in ls and press the enter key
You should see a list of users for your Mac.
Now, if we assume your username is Marvin, type in cd marvin (note there is a space after 'cd'). Obviously, if your username is something else, then use that instead of marvin!
Press the enter key
Now type in ls and press the enter key
You should see a number of folders listed. Type in cd downloads (note there is a space after 'cd') and press the enter key
Now type in ls and press the enter key
You should see a list of files and folders. Now use the rm command to remove items you don't want.
For example, if you have an item called Installer.dmg, you would remove it by typing in rm Installer.dmg (note there is a space after 'rm') and pressing the enter key
Once you've removed enough items to free up space, reboot your Mac by typing in reboot and then pressing the enter key


Note: If an item has a space in its name (e.g. Install XYZ.dmg), then the command you would need to enter is rm Install\ XYZ.app. That is, I have used a backslash \ before the space!

Deleting Applications via Single User Mode

If you want to remove applications, then follow these steps:


Follow Steps 1 to 5 from Deleting your sleepimage file via Single User Mode above
Now type in cd applications (note there is a space after 'cd') and press the enter key
Now type in ls and press the enter key
You should see a list of applications and folders. Now use the rm command to remove items you don't want.
For example, if you have an item called XYZ.app, you would remove it by typing in rm XYZ.app (note there is a space after 'rm') and pressing the enter key
Once you've removed enough items to free up space, reboot your Mac by typing in reboot and then pressing the enter key


Note: If an item has a space in its name (e.g. Microsoft Word.app), then the command you would need to enter is rm Microsoft\ Word.app. That is, I have used a backslash \ before the space!