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

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

Create a new admin account

1. Hold ⌘+S on startup
2. mount -uw / (fsck -fy is not needed)
3. rm /var/db/.AppleSetupDone
4. reboot
5. Go through the steps of creating a new account. You can leave all fields in the personal information step blank.
6. After logging in to the new account, go to the Users & Groups preference pane
7. 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)

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

Using iCloud Recovery Mode (10.9.x & newer)

1. 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"
2. Click the arrow button to bring up a "Reset Password" dialog
3. Enter the Apple ID login information that is tied to the OS X user account and click “Reset Password”
4. Click “OK” to confirm the new keychain creation
5. Enter and verify a new password, fill out the password hint field, and again click on “Reset Password”
6. Click “Continue Log In” to login as the user account with the reset password

There are at least three ways to reset the login password of an account.

Using Reset Password.app on the recovery partition:

1. Hold command-R on startup.
2. Select Utilities > Terminal.
3. Run resetpassword, which opens the Reset Password application.
4. Select the volume and account and reset the password.

Using dscl in single user mode:

1. Hold command-S on startup.
2. Run mount -uw /.
3. Run launchctl load /System/Library/LaunchDaemons/com.apple.opendirectoryd.plist in 10.7 or later or launchctl load /System/Library/LaunchDaemons/com.apple.DirectoryServices.plist in 10.6 or earlier.
4. Run dscl . passwd /Users/username password where username is the name of the account and password is the new password.
5. Run reboot.

Creating a new admin account:

1. Hold command-S on startup.
2. Run mount -uw /.
3. Run rm /var/db/.AppleSetupDone.
4. Run reboot.
5. Go through the steps of creating a new account.
6. Reset the password of the old account from the Users & Groups preference pane.

If FileVault 2 is enabled, you need to enter a password to start up in single user mode, and the list of accounts in Reset Password.app will be empty, so none of these methods can be used.

None of the methods above resets the password of the login keychain. If automatic login is enabled, you can print the password of the login keychain by running a command like this in single user mode before resetting the login password:

sudo ruby -e 'key=[125,137,82,35,210,188,221,234,163,185,31];IO.read("/etc/kcpassword").bytes.each_with_index{|b,i|break if key.include?(b);print [b^key[i%key.size]].pack("U*")}'