YouTube's comment system works primarily through Google Plus' notification system instead of your inbox, but your inbox still exists and can be used to contact other YouTube users. Comments from videos no longer show up there, but you can use it to view and respond to any personal messages.

Check Your Messages

Make sure you are logged in to your Google account, then open YouTube's home page. Click your email address by your profile picture, then click "Video Manager." Click "Inbox" from the sidebar, then select any of the subcategories of messages you want to view. While comments are no longer delivered to your inbox, all of your old messages will still be in your inbox unless you deleted them.

Message Other Users

You can compose a message from your inbox to anyone in your contact list, but you can write a message to any YouTube user via his YouTube profile. Open a user's YouTube channel (for example, youtube.com/username), then click "About." Below the channel summary, click "Send message." This opens the message composition window with the user's name filled in. Sending a user a private message does not add that user to your contacts, however, so you will need to follow this same process to send him a new message if you wish to do so again.