There are two basic types of email services. POP3 and IMAP.
POP3 is the most basic type. Emails are stored on a server only until you retrieve them with your computer (or phone). When you check your email, the emails are physically downloaded off the server, into your computer, and then deleted from the server. This arrangement works well if you only check your email using the same device every time. This type of service is typically free, as you are responsible for keeping the downloaded emails safe on your computer. The downside of this arrangement is that the emails cannot be synced among multiple devices.
IMAP email is stored on a server. When you check your email, whatever device you are using simply displays what is on the server. The emails are left on the server until you manually delete them. When you delete, mark read/unread, etc on one device, the change is made on the server, and immediately reflected on all other devices that are viewing the same set of emails. This is obviously a superior arrangement, however, it requires a large amount of storage space on the server to maintain everyone's emails, so it's not typically offered for free. (Gmail is the exception.)
Gmail can be configured as either POP3 or IMAP. Sounds like your setup as POP3 and want to be setup as IMAP.
Unfortunately, this won't be an option with Hotmail...
If syncing the hotmail account is an absolute must, there are services which can basically convert a POP3 account to an IMAP account for about $5 a month, such as
Mobile Email & Push Email Services | mail2web.com