The default behavior for conversation routing in Live Chat Manager is to route each new conversation to the best available agent. Best available means the available agent with the least number of open conversations. There are scenarios where this is not a desirable routing method, and where you would prefer the conversation be routed to a specific agent.
For SMS Chat, the way to route conversations to a specific agent is to ensure the SMS Sender is associated with the Agent. Go to Administration » Users » Edit and set the Assigned Texting Phone Number for the agent.
If multiple agents are assigned the same Assigned Texting Phone Number, it would then select the best available agent from that group of agents.
Once a conversation has been assigned to an agent, and for as long as that conversation remains open, all subsequent messages coming from the user will be routed to that agent.