Let's take a look at the most common scenarios:
You give your password to a friend or family member
Nooooo! Don't do it! The only person that should know your password is you. Don't forget that accounts are personal. Each summoner is unique, which means that when his account is shared, part of its personality is lost in the process. Don't give your password to that friend who wants to check out your skins or to that cousin who just joined the League and wants to know what a level 30 account feels like. You can always help them create their own account. That way, you can even start a team and play together.
Free RP offers
Don't take the bait. There are shameless people out there who want to take advantage of innocent players by offering free or discounted RP. They will ask for your password in order to transfer the RP. And you know what happens when you give it to them? They keep your account and you never see the RP. The only way to purchase RP is through the in game store or with the pre-paid cards sold at stores and cybercafes. If you see anyone offering RP on another site, don't trust them, and submit a request below reporting the abuse. Piltover will be grateful!
As we said before, accounts are personal, non-transferable and free. Both selling and buying them is strictly prohibited. Not only is it bad practice, it is also often used to take advantage of players. Someone sells you their account, keeps your money and then recovers said account. You end up with no account and no money. Don't buy or sell accounts. In both cases, you lose.