First of all, let's get the names sorted.....
Silver solder is an alloy of metals designed to have as close a colour match to silver as possible, but with a lower melting temperature than silver itself - you want the solder to melt well before your work does! The photo above shows Hard, Medium and Easy solder strip, usually the most common way that silver solder is sold in. There are also a couple of syringes of solder paste in the background, and I'll come back to those later.
Hard, Medium and Easy simply refer to how hard or easy the solder is to melt. Hard has a higher melting temperature than medium (ie. it is harder to melt), medium has a higher melting temperature than easy, which is the easiest to melt! The different types of solder all form nice strong joins if used properly - the names refer to how hard or easy they are to melt, not how hard or strong the join is.
By the way, you can also buy enamelling solder that has an even higher melting temperature so that your hard work doesn't fall apart when you pop it in the enamalling kiln, and extra-easy solder which has an even lower melting temperature than easy solder. I find that extra-easy solder is a bit sluggish to flow and so only use it if I really have to.
But why do you need different melting temperatures?
You will often need to solder more than one join on a piece of work, and when you solder the second join, you don’t want the first to remelt! Working down through the melting temperatures of solders for each new join helps is one of the easiest ways to protect the previous joins.