Well the answer is yes, of course you should. But there maybe some extra costs involved as you may need a certificate. You will need to talk to your hosting company to see what they are offering.
So what does this HTTPS mean? Well if set up correctly it means all data to and from the website is encrypted. If you didn't already know if you're accessing the internet on some public wifi connections it is quite easy to intercept your connection and see what you are doing. So entering passwords, filling in forms or whatever could all be seen by persons unknown. On a Secure website as this data is encrypted it would be much harder to see or track what you are doing on the website.
Google are starting to flag up 'not secure' websites. A warning that the website is not secure is displayed on websites that haven't got HTTPS. Currently this appears to be limiuted to websites that take secure information such as credit card details, though it could be a login with a password.
Eventually Google apparently wants all websites to be secure, so you have been warned.