Hi, Laura!
It's great to hear you're liking osTicket so far.
Mcrypt is an encryption library for PHP. It has to be installed by your server administrator/web host.
With regards to its function in osTicket, when you set up your e-mail accounts in osTicket for e-mail fetching and sending, you have to enter the password for that account. When you do so, it gets stored in the database. If the mcrypt extension is enabled, then those passwords are encrypted before being stored in the database, making them more secure, helping protect your data in the (typically unlikely) event that the server is ever compromised.
osTicket will work just fine without mcrypt, but those passwords won't be encrypted. - It's a good idea, but not required.
If your host can't/won't install mcrypt for you, then my recommendation would be to just ensure you aren't storing any passwords for personal email accounts and that the email addresses you've configured have unique passwords that aren't used elsewhere.
