I suggest booking directly with www.trenitalia.com rather than through third parties.
Economy and super economy tickets are simply types of advance purchase tickets that are cheaper than base fare. For example, a super economy ticket in business class will be cheaper than an economy ticket in business class which will be cheaper than a base ticket in business class. Tickets for the high speed services you are interested in can be booked up to 120 days in advance and supereconomy and economy tickets can sell out quickly.
For Super Economy, refunds, change and access to a different train are not permitted.
With economy tickets you can change date and hour once up to departure, choosing a journey on the same category of train, according to the ticket change procedure and paying the difference between the full price (Base) applied to the train used and the sum already paid. Refunds are not possible.
With a base ticket, before departure you can change your base ticket free of charge and unlimited times; after departure, up to 1 hour, free of charge once at train ticket office at the departure station only. If you no longer intend to use the ticket and want to cancel the journey, you can request a refund before departure with a deduction charge of 20%. After departure refund is not allowed.
As well as choosing a ticket type, you need to choose a class. Seat 61 gives a great summary of the differences between the classes on Italian trains, including showing photos. Generally, I just travel in the cheapest class available, although I have traveled in business when I have got supereconomy tickets. For example, on a frecciarossa train I would book standard class (higher classes are premium, business and executive)