To join the public chat, hover over your username in the top right, and under the Community section on the left, click "Chatroom". To use the chat, enter your message in the white box below the main chat screen and press ENTER. Your nickname will appear in the screen, along with your message.
To start a private chat with someone, go to that user's profile and press the grey "Chat" button. This will open a chat window, and send them an invitation to join.
If you have any incoming invitations, you will receive a notification showing which member has invited you. You can either click "Join" to enter the chat, or "Decline" to decline the invitation.