Saving throws are used when an effect calls for a saving throw, eg. fireball which requires a dexterity saving throw. In this case, there is a set DC which the target of the spell or effect must roll over in order to avoid or soften it. Each class is proficient in two saving throws. A saving throw is your ability modifier unless you're proficient, then it's mod+prof.

A skill check refers to all the skills like athletic, handle animal, history, insight, bluff and they are used when a character is attempting to do something and needs to roll for success. In this case, you roll a d20 + skill's base stat modifier (Str for athletics, dex for stealth, wis for insight etc) + prof if you have proficiency in that skill. In this case, the character attempting to do something rolls against a preset DC made by the DM to succeed.

Ability checks refer to attacks, spells, stat contests and so on where the target has to roll as opposed to having a preset DC. If you want to cast a fireball, you won't roll, your target will roll a saving throw. If you, on the other hand, cast a spell that requires you to make a "spell attack" or try to attack someone with your weapon, you will roll your d20 against the enemies AC. If you're attacking with a weapon, this would be d20+str or dex mod+ prof if you're proficient in that weapon. If you're attacking with a spell this would be d20+your spellcasting ability mod (int for a wizard, cha for sorcerer and warlock, wis for cleric etc)