Follow the below steps to create a new mailbox in Exchange Server 2013 :

1. Login into Exchange Admin Center ( EAC ) . read this article if you don't know how : How To Access Exchange Admin Center in Exchange Server 2013  
   
   
2. In EAC, the first page you will presented with is the recipients page.
   
   
   
   Under mailboxes, click the + button and then click User mailbox
   
   
   
   
   
3. Now you will need to fill all the information and then click save
   
   
   
   
   In the Alias box, type the user’s alias. It must be unique in the forest. The user’s alias is the portion of the email address on the left side of the at @ symbol.
   
   First name : Type the first name of the user.
   
   Initials :  Type the initials of the user.
   
   Last name : Type the last name of the user.
   
   Display name :Type a display name for the user. This is the name that's listed in the mailbox list in the EAC and in the shared address book. By default, this box is populated with the names you enter in the First name, Initials, and Last name boxes. If you didn't use those boxes, you must still type a name in this box because it’s required. Note : The name can't exceed 64 characters.
   
   Name :Type a name for the user. This is the name that's listed in Active Directory. This box is also populated with the names you enter in the First name, Initials, and Last name boxes. If you didn't use those boxes, you must still type a name because this box is required.
   Note : This name also can't exceed 64 characters.
   
   Organizational unit : You can select an organizational unit (OU) other than the default (which is the recipient scope). If the recipient scope is set to the forest, the default value is set to the Users container in the Active Directory domain that contains the computer on which the EAC is running. If the recipient scope is set to a specific domain, the Users container in that domain is selected by default. If the recipient scope is set to a specific OU, that OU is selected by default.
   To select a different OU, click Browse. The dialog box displays all OUs in the forest that are within the specified scope. Select the desired OU, and then click OK.
   
   User logon name : Type the name that the user will use to sign in to the mailbox and to log on to the domain. The user logon name consists of a user name on the left side of the at (@) symbol and a suffix on the right side. Typically, the suffix is the domain name in which the user account resides. Note that you can’t use an apostrophe (') or a quotation mark (") in the user logon name because these characters aren’t supported.
   
   Confirm password : Use this box to confirm the password that you typed in the Password box.
   
   Require password change on next logon : Select this check box if you want the user to reset the password when they first sign in to the mailbox.
   If you select this check box, at first sign-in, the new user will be prompted with a dialog box in which to change the password. The user won't be allowed to perform any tasks until the password is successfully changed.
   
   
   Once all information is filled, click save
   
   
   
   
   
4. The new user mailbox will be listed as displayed below
   
   
   
   

Now you can logon to the newly created user mailbox

1. Open Outlook Web App (OWA), and sign in with the new user credentials
   
   
   
   
2. Choose your preferred display language and time zone , then click save
   
   
   
   
   
3. Done, the user is logged into his/her new mailbox
   
   
   
   
   

Summary

Creating a new mailbox in Exchange Server 2013 is an easy step, as if you are creating an outlook.com email address.