There are many techniques for keeping your Outlook information on two different machines. However, Microsoft has not released a synchronization utility for Outlook. Their solution is to use an Outlook.com, Office 365 or an on-prem Exchange mailbox.
The methods on this page or sync utilities are not necessary if you have an Exchange account (including Outlook.com) as it syncs messages, calendar and contacts between Outlook and online. Phones and tablets will sync with the server.
For simple "syncing" of email, choose an IMAP email account if your email provider supports it. If you use a POP3 account, leave mail on the server until the other computer has downloaded the messages and BCC yourself on sent messages. You can send appointments and contacts to yourself as needed, or use one of the methods below to copy non-mail items to your other computers.
To copy calendar and contacts or archived mail to a second computer, copy the PST data file between your computers. You can do it the old-fashioned way, using a "sneaker net" and transfer the pst on a USB drive, or access the other computer over your local network and copy the PST. If you use an online service, such as Dropbox, OneDrive etc, to transfer the PST file, you should copy the PST to a folder outside of the sync service. The PST can't sync while Outlook has it open but the service will keep trying, often creating a lot of small files which waste your storage space. This can corrupt the PST.
If you want an automated method, you'll need to use one of the tools listed below.
For tools you can use to sync between Exchange Public folders and mailboxes, see Synchronizing Exchange Mailboxes and Public Folders.
Synchronizing Outlook using Web Services has information on using Internet services to sync two copies of Outlook.