Hi Tracy,
In that case, you'll just want to make sure your agent-group relationship is set accordingly. Go to Manage > Groups and edit the group. On the Agents tab, double-click your username and make sure the Round Robin option is set to Not Included (Pull from Queue). In order for a ticket to be assigned to you, you'll need to Take it from the Queue or another agent. No tickets will be assigned to you automatically.
If the other agent in that group is set to Primary Group for the Round Robin option, then all new tickets will go to them by default. If they go inactive from that Group using the Active Groups window, then tickets will start going to the Queue instead, where one of you can Take them as needed.
You can also utilize the Max Tickets setting to make sure that agent is not overwhelmed with tickets. Assuming you two are the only agents in the group... If you have your Round Robin set to Not Included (Pull from Queue), then tickets that go over the agent's max limit will be sent to the Queue, where either of you can Take them. On the other hand, if you have your Round Robin option set to Secondary Group, then tickets that go beyond that agent's max limit will be assigned to you by default instead.