Your tickets have different statuses depending on in which state it is.

A ticket which no one has responded to, awaiting a reply.

A response has been sent by an agent awaiting a response from the requester.

A response has arrived from the requester, waiting to be read and responded to.

A status set manually by the agent. It can be used as a kind of 'on hold' mode where you have solved what you can solve now, but where you wait for a response or a solution from a third party before you can answer the customer again. You can also set the status when you are not really sure if the case is completely resolved, but where you have done what needs to be done.

Should the requester send a reply when the ticket is set as Solved, the ticket is reactivated och set to Reply. This status should be used if the agent wishes to retain the historical feed of the earlier messages and tickets.

Is used to close a ticket. If the requester answers the same ticket, even though it is closed, the history will be included, but a new ticked ID will then be created for this. If the requester sends a completely new email, it will be a new ticked with no previous history. The requester will not be notified if the ticket status is changed to closed. 

You can find more information here: What's the differens between Solved and Closed in Tickets?