Imagine you're a european in the middle ages and you're defending a castle against a siege. You don't think you're going to win - you think you'll probably be starved then assaulted and everyone will be killed or captured. So you start to negotiate with the people outside.

"If we surrender the castle, will you let the defenders go free and unharmed?"
"We will let the defenders go free as long as they surrender their arms and leave Lord Gloucester's son as a hostage."

This is a simple form of conditional surrender. It was a standard way for castles to be taken - they'd hold out for some length of time and then come to terms. Of course, negotiating terms is of limited use unless you can trust the victors - for example they fear the judgement of god or other nations.

If the surrender was unconditional, that literally means the victor has no conditions imposed on them. For example, there is no agreement not to kill the surrendering soldiers, something which was not uncommon in medieval warfare.

The same principle applies to armies or countries. Countries will want to try and guarantee their territory, institutions, governments, treaties, etc..

For example, a sticking point for Japan's surrender in WW2 was the status of the Emperor. Although there was a consensus in the Japanese leadership that the war couldn't be won, to surrender and have the Emperor dethroned, or even tried and executed as a war criminal was something they couldn't accept. Japan did surrender unconditionally, but only after both the devastation of the atomic bombings and some reassurance from the US over the issue of the Emperor.

I've seen it argued that the insistence on unconditional surrender was an unusual feature of 20th century warfare – or perhaps American or democratic warfare. Previous conflicts were more likely to end with concessions and conditions, which might have ended wars earlier, since countries did not have to be pushed to the brink in order to accept unconditional surrender.