Q: How can I disable the password

To stop the computer from asking for a login password:

Click on the Apple > System Preferences > Users and Groups > Login Options

You'll have to click the lock and type in your password to make changes. From here, select the drop down menu that says "Automatic Login," and select your account. Type your password again and click ok, then click the lock again to lock it.

As for needing your password to download things, that is a normal function put in place to make sure no one else can download potentially harmful things on your computer.

You can partially change these settings in Apple > System Preferences > App Store

Under Password Settings, you can change it to save your password for all FREE downloads, but you can only set it to keep your password for 15 minutes on purchases and in-app purchases.

1.  To disable the need to login with your password when starting up your Mac, go to System Preferences --> Security & Privacy.  Click the lock at the bottom and input your password, then make sure the top three boxes are not checked.  After you're done, click that lock again.

2. The reason your computer asks for your password when you're downloading programs onto your computer is it is making sure that the person doing the downloading is the person who the computer belongs to.  It's a safety feature that to my knowledge cannot be turned off, nor should it be.  Otherwise, what would stop someone from getting a hold of computer, even a friend or family member, and putting something on there without your knowledge?  If they don't know your password, and they shouldn't, they wouldn't get very far.  I had to input my password just to update my Adobe Reader.  Is it annoying?  I don't know if I'd go that far. 

Just to clarify...

You will need to do both answers posted to ensure it never asks for a password while logging in. Pinkstones' method will prevent the computer from asking for a password when it wakes from sleep, my method will allow your computer to go straight to the desktop when the computer is restarted or turned on. My method also automatically changes the "Disable automatic login" setting in Security and Privacy.

You need to go to System Preferences->Users&Groups, unlock, and on the left,

click the item at the bottom saying Login Options.  Then in the drop down menu

select the automatic login user.

You are trying to set the global option in Security when it hasn't been set for a user.

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Could be a permissions issue. This can take a long time to complete, so be patient and know the computer will run slow while doing this. I would try to just not use it while this runs.

Repair disk permissions with Disk Utility - Apple Support

I think you can get around this problem by creating a new Admin user, and then logging into the new user to change the settings for your current user.

Here's how it's done:

1. Go to Users and Groups again, unlock the lock
2. Click the +
3. Next to "New Account," select Administrator, then create a really easy password like "test" and create the account
4. Once it's created, click the lock to save and then completely log out of this user
   1. Apple > Log Out (Your Name)
5. Log into the new account and run through the basic set ups without using iCloud or anything like that
6. Run through the same steps I posted earlier, and see if you can select your original user under Automatic Login
7. If it works, just log back into your original account and delete the test Admin account by going back to Users and Groups, and hitting the
   1. You can delete the home folder of the test account