HostGator offers you a choice in webmail programs. You can pick whichever one you are most familiar with, or the one you learn to like over time.
You can switch webmail programs as much as you like, and you will still see all your emails.
Horde is a popular choice, but does require some configuration to make it behave the way users expect. (For instance, Horde does not use a trash can for your deleted emails.)
RoundCube is another popular choice. Again, just pick the program you like.
Don't forget, you can change webmail programs anytime, and you will still see the same emails.
How to Use Webmail
To help you become familiar with webmail, feel free to watch the following video: