Saving throw is a blanket term for avoiding a negative effect either through the environment, enemy abilities, manipulations, or items. Many things in the world will trigger saving throws of various types and the character's capacity to resist these effects will in large part determine their fate.
There are three saving throws, one each for Earth, Air, and Water. A saving throw is made when a character is called on to resist a negative affliction of some kind. A player may choose to fail a saving throw on purpose.
In order to make a saving throw, a character rolls a d20 and adds the appropriate modifier depending on the affliction. This can be a single element, such as an Earth saving throw, or a combination of elements.
Some abilities, poisons, or element effects afflict characters in different ways and may require multiple saves as listed in an ability's description or at the discretion of the GM. For instance, a weaker ability may force a character make either an Earth or Air save, whichever they prefer, or roll both and if either is passed the saving throw is considered passed. Other more powerful abilities may require two or even all three element saving throws to be passed to avoid the negative effect. The purpose of this is to create flexibility in the gameplay while maintaining a set of only three saves for a player to track.
Many character abilities will state something to the effect of "target must make an Earth saving throw or..." This saving throw is calculated as 5 + Element modifier x 2. In keeping with the above example, if an enemy must make an Earth saving throw against a Soldier's ability and that Soldier has an Earth modifier of 4, then the enemy must meet or exceed 13 (5 + (4 x 2)) in order to make the saving throw and avoid the negative effect.
Ability checks are d20 rolls made when a character attempts to do something that is not effortless. All are usually accompanied by an appropriate Element modifier and always by a target DC that must be met or exceeded. Trying to hit an enemy in combat is an ability check and so is making a leap across a gap in a broken bridge. Even saving throws are technically ability checks that test whether a character can shrug off a negative effect or not. In short, every d20 roll in the game is an ability check of some kind.
This book has no list of skills linked to particular attributes. This is purposeful and allows for tasks to be approached from different perspectives. The way a character decides to tackle a problem could change which Element modifier is used. What I recommend is for the players and GM to decide on the essence of a task and use the modifier(s) that fit best. I will give a few examples to illustrate.
A character wants to jump across a gap in a broken bridge. This is always an Earth check excepting some extraordinary circumstance. Why? Could it not be Air? Air makes you lighter and it kind of feels like dexterity. Or Water, maybe? You really need to make this jump or others will suffer. That counts, right? No. You jump with your legs, not with your lightness or need. At the GM's discretion, the other factors could modify the target DC, but jumping is about bodily strength and coordination.
Attacking (specifically "to hit") and Defense both are based on the Dominant Element modifier because they are two sides of the same coin. That is the coin of how one fights. It makes no sense to attack with one element in your makeup and defend with another. The truth is they are all connected, but one rises to the fore, especially in desperate, life and death situations. Training can overcome these instincts or enhance them, hence why the highest Element modifier is always your Dominant Element.
The last example is attempting to intimidate (a form of persuasion) an NPC for some purpose. This could be any of the elements depending on the context and how the character approaches it. Will they use physical presence and potentially violence? Sounds like Earth to me. How about arcane knowledge or sarcasm to tear someone down? Air. What if they take an indirect route and work a room, making friends with everyone and forcing someone's hand by the power of their social credibility? I'd buy that argument for Water.
The point of these examples is to find the essence of a task and to show that some tasks have clear solutions, some are connected, and some very much depend on how they are attempted.