How can I get admin access to a Mac without knowing the current password?
With macOS Catalina and many recent macOS - the built in tools guide everyone how to reset admin permissions without needing any of the single user mode boot tricks of old (left at the bottom for posterity and people running OS that were shipped before 2015)

Now - follow the official Apple triage steps: https://support.apple.com/en-us/HT202860

Instead of single user mode, boot to recovery and use the Terminal Utility. Then you can remove the .AppleSetupDone file from:

/Volumes/Macintosh\ HD\ -\ Data/private/var/db 




There are many ways to reset the login password without installation media, here are three: 

Change the password in single user mode


Hold ⌘+S on startup
mount -uw / (fsck -fy is not needed)
launchctl load /System/Library/LaunchDaemons/com.apple.opendirectoryd.plist (or /System/Library/LaunchDaemons/com.apple.DirectoryServices.plist in 10.6)
dscl . passwd /Users/username (without a trailing slash) and enter a new password. You can ignore the error about com.apple.DirectoryServices.plist.
reboot


Create a new admin account


Hold ⌘+S on startup
mount -uw / (fsck -fy is not needed)
rm /var/db/.AppleSetupDone
reboot
Go through the steps of creating a new account. You can leave all fields in the personal information step blank.
After logging in to the new account, go to the Users &amp; Groups preference pane
Select the old account, press the Reset Password... button, and enter a new password


Restart and enter Recovery mode (for 10.7 Lion and newer OS only)


Hold ⌘+R on startup
Open Terminal from the Utilities menu
Type resetpassword and follow the instructions


Using iCloud Recovery Mode (10.9.x &amp; newer)


After incorrectly entering a user password three times at the Mac login screen, a message appears saying "If you forgot your password, you can reset it using your  Apple ID"
Click the arrow button to bring up a "Reset Password" dialog
Enter the Apple ID login information that is tied to the OS X user account and click “Reset Password”
Click “OK” to confirm the new keychain creation
Enter and verify a new password, fill out the password hint field, and again click on “Reset Password”
Click “Continue Log In” to login as the user account with the reset password