To log out of League of Legends, click the X in the top-right corner of the League client. A dialog appears with two buttons: Exit and Sign Out. Click Sign Out. Exit just closes the window without logging you out.

Signing out returns you to the Riot login screen. There's no account-switcher in the current client; sign out and sign back in with the other account.
Logging Out of the Riot Games Launcher
Clicking X on the Riot Games launcher doesn't log you out. It minimizes to the Windows system tray and keeps running in the background. To fully exit and log out, click the arrow icon in the taskbar (bottom-right, near the clock) to reveal hidden tray icons, right-click the Riot Games icon, and select Exit. The next time you open the launcher you'll be taken to the login screen.

If your account may be active on another device, go to account.riotgames.com and navigate to Security to terminate all active sessions remotely.
If logging out didn't resolve your issue, you may need to repair your League of Legends installation instead.